a145a24a89 a135a73a166a232a73
2001a3510a21911a134
1 a135a232a73a123a229a205,a58a238a166a116a89a26a251
a89a116a135a232a73,a33a7a155a123a220a199a124a52a58a238(Issac Newton,1642-1727)a203a116a89a26
a251(Gottpied Leibniz,1646-1716),a135a232a73a210a52a198a162a21a25a123,a7a149a58a238,a139a108
a85a123a52a198a41a89a199a211a42a52a243a166a9a123a30a127,a15a142a7a28a162a123a181a106a164a7a66,a0a199a30a127,
a15a210a5217a45a22a0a199a30a127,a78a179a166a60a105a184a51,a78a179a139a234a105a245a124,a198a243a93a41a74a24
a76a166,a79a143a166a60a123a7a248,a166a68a56a34a234,a198a210a195a27a243a27a176a41a7a149a135a232a73a123a89a74
a211a42,a116a89a26a251a52a24a199a200a48a193a209a58a158a128a132a123a124,a106a166a52a198a123a108a85a123a24a92a73,
a198a123a108a85a116a13a115a1a27a70a200a48a193a209a139,a198a162a220a124a131a45a139a154a112a88,a52a79a143a112a88a120
a52a135a232a73a123a21a25a86,a89a199a112a88a52a88a115a123,a15a150a139a31a143a63a66,a34a159a222a52a116a89a26
a251a62a24a199a21a44a7a149a135a232a73a88a169a123a124,a198a123a88a169a2431684a35a21a44,a58a238a41a89a199a211
a42a0a149a116a89a26a251,a13a116a89a26a251a21a44a88a169a0a149a58a238,a58a238a139a234a89a199a211a42a128a150a139
a21a44a31a143a127a91a123a192a220,a13a116a89a26a251a88a98a21a44a234a89a74a192a220,a51a30a164a90a126a234a24a74
a110a82,a15a142a183a162a25a243a164a243a126,a0a143a128a117a220a199a124a139a24a199a112a88,a76a150a209a52a203a106a166
a150a139a7a248a123a124a243a0a176a5a196a123a60a89,a89a88a52a24a199a31a143a139a63a135a123a47a60.
2 a135a232a73a228a253a189a174
a135a232a73a52a106a166a176a62a105a173a7a123a48a193,a150a149a31a143a52a135a232a73a17a218a183a46a135a73a123a21a25
a203a136a11a14a21a25a43a41a58a158a139a7a248,a79a143a136a11a14a21a25a43a41a131a128,a106a166a204a7a123a248
a123a210a52a207a196a60a106,a207a196a220a28a106a123a7a248,a139a171a171a123a7a248,a183a162a127a119,a60a106a139a171
a171,a139a34a117a123,a58a34a117a123,a174a110a60a106a127a171a171a60a106,a0a143a7a14a143a248a143a207a196a60a106a123
a117a159? a183a162a209a127a119,a60a106a44a49a126a77a34a117a44a43,a152y = f(x) a89a30a77a34,a243a89a30
a77a34a123a154a154,a152a42a199a52a44a49a135a73a123a155,a0a143a199a243a154a154a139a199a35a34,a135a73a210a52a178
a89a199a77a34a126a199a123a35a34a117a207a196a199a123a117a159,a196a49a15a127a149a128a199a52a178a60a106a34a117a154,a34
1
a117a154a131a128,a44a49a28a1a62a1a198a1a248,a44a49a15a174,a79a36a44a49a122a116a106a241a117,a106a166a7
a52a21a234a122a116a106a241a117,a15a52a105a7a155a123,a196a49a135a73a76a150a52a128a126a77a34a123a35a34a117a207
a196a77a34a123a117a159,a232a73a117a122a0a234,a79a143a232a73a34a19a222a76a151a89a229a117,a199a52a7a15a174a193
a232,a0a143a34a38a178a193a222a139a24a199a75a173,a132a77a34a117a41a143a30a142,a199a123a193a232a139a245a76,a203a177
a123a193a232a139a245a76,a89a176a123a175a11a52a232a73a123a13a41,a15a52a232a73a173a7a123a248a123,a79a36,a34
a19a222,a232a73a123a21a48a243a135a73a131a3,a232a73a104a30a15a150a139a24a189a123a189a66,a232a73a210a52a139a199
a244a33a123a10a39,a77a34a196a140a194a123a75a173a24a196a46a27a6a126a134a34a117a5a163,a38a122a5a163a123a77
a34a74a149a28a123a30a142a123a30a127,a210a139a199a244a33,a210a52a89a199a75a173a123a193a232,a196a49,a15a13a211
a131,a232a73a123a21a48a243a135a73a131a3,a165a45a89a220a199a175a11a80a54a150a139a7a248,a98a52a217a34a89a7
a248a58a158a123a183a35,a232a73a139a88a245a52a135a73a123a39a228a174,a7a48a128,a34a38a28a70a89a30a134a34a203
a220a30a18a34a196a196a75a173a123a193a232,a89a220a30a18a34,a24a199a52s = a,a24a199a52s = x,a28a7a86
a174a89a199a75a173a123a193a232,a52a199a189a232a73integraltextxa f(x)dx,(a214a42f(x)a189a232a73a44a → x),a89
a52a104a35a116a89a26a251a123a110a82,a89a199a232a73a123a110a82a16a196a89a248,a79a143a232a73a15a52a83a44a24
a199a90,integraltexta83a44a90(sum),a34a247a193a232a24a30a132s = a a123a134a34a42a30a142,a12a24a30a52a127a63
a123x,a28a178xa89a30a134a34a35a196a123a155,a210a122a116a24a199xa123a60a106,a89a199a60a106,a183a119a199A(x),
a210a52a183a67a222a123a193a232,a52a199a232a73,a196a49a199a52a24a199a106a248,a166xa139a7,a196a49a52xa123a60
a106,a89a199a60a106a203a77a34a48a199y = f(x)a89a199a60a106a139a183a35a7a248,a143a31a143a139a183a35a123a7
a248a17a218a105a59a92a143a23a23,a34a152a70A(x) = integraltextxa f(x)dxa123a135a73,a70a199a123a135a73a108,a210a52
a128,a70s = x,x + δx a196a140a196a123a89a199a66a30a75a173a123a193a232,a25a243a152a42a28a252δxa248a123a155,
a183a46a105a142a52a23a241a117a234,a89a199a244a33a210a52f(x),a196a49a105a142a52a23a241a117A(x)a89a199a60a106
a123a135a73a210a52f(x),a79a36
dA(x)
dx = f(x),(1.1)
a89a210a52a135a73a51a232a73a123a228a253a123a7a248,a89a199a7a248a128A(x)a52a24a199a232a73,a70a199a123a135a73
a123a30a127,a210a122f(x),a89a199a24a196a143,a119a41a135a232a73a123a228a253a189a174,a183a44a3a243a8a13a39a135
a232a73a123a30a127,a41a170a88a195a143a31a143a89a52a24a199a135a232a73a123a228a253a189a174,a79a143a24a196a143a178a89
a199a7a248a42a85a196 integraldisplay
f(x)dx|ba =
integraldisplay b
a
f(x)dx (1.2)
a111a231,a38a30a232a73a52a199a88a189a232a73(indefinite integral),a88a189a232a73a52a199a60a106,a38a42
2
a52a60a106a243ba123a138a62a86a60a106a243aa123a138,a127a149a89a199a189a232a73(definite integral),a196a49
a44a89a199a7a248a127a119a7a70a232a73a123a155,a144a137a7a70a24a199a60a106,a199a123a135a73a52a46a127a123,a210
a52f(x),a253a135a73a52a46a127a123,a196a49a89a248a135a73a203a232a73a203a229a117a234,a144a35a123,a232a73a127
a149a135a73a123a39a228a174,a139a234f(x),a7a73a24a199a60a106,a199a123a135a73a127a149f(x),a52a199a39a228a174.
a79a36a135a1a232a73a139a183a35a123a7a248.
3 a245a195a135a232a73
a222a193a89a123a52a24a199a35a106a123a135a232a73,a5a193a89a176a145a123,a7a245a35a106a123,a245a35a106a123a155,
a139a99a123a25a54,a52a31a143a248a123a17a218a183a46a233a149a245a35a106a123,a183a162a11a88a23a47a123,a11a23a220
a199a35a106a123a60a111,xa203y,a0a143a183a162a127a119a89a199a30a127a135a73a123a10a39a123a77a18a52a160a135a73,
a127a149xa203ya73a13a70a135a73,a232a73a123a10a39a77a18a52a173a232a73,a19a173a232a73(double integral)
a52a2432a145a123a60a111,a243a176a145a123a60a111a52a245a173a123,a11a232a145,2a145a123a60a111a210a139a234a75a173,
a183a162a119a199?,a0a143a199a123a30a142a119a199γ,a196a49a232a73a123a24a199a7a108a77a18a52a24a1992a173a232a73,
a202a47a232a73a178xa73a196a66a227,a108a128a82a66a227a242a198a222a89a199a60a106,a70a24a199a90,a2432a173a232a73
a123a30a127,a48a27a15a52a178a75a173a73a196a66a76,a108a128a82a154a24a66a76a123a193a232,a243a217a222a60a106a138
a198a222a199a123a193a232,a108a128a70a199a123a90,a105a88a122a234a123,a34a38a60a106a80a123a155,a195a88a28a152a91a87
a28a123a75a173,a244a33a52a24a248a123,a196a49a89a244a33a210a522a173a232a73
I =
integraldisplay integraldisplay
f(x,y)dxdy,(1.3)
a2432a145a123a30a127,a4a150a176a145a123a30a127,a24a199a173a7a123a25a54a52,a183a162a25a243a1392a199a35a106x,y,
a166a35a106a14a143a248a218a196a49a183a25a243a166a35a106,a166a35a106a104a108a52a243a135a232a73a176a52a105a173a7a123
a24a199a205a27,a79a143a105a245a123a175a11a52a23a28a123a35a106a52a100a160a10a122a55a104,a139a30a166a35a106,a175
a11a210a193a47a59a92a154a234,a210a44a49a137a251a234,a25a243a183a166a35a106a213
braceleftBiggx = x(xprime,yprime)
y = y(xprime,yprime) (1.4)
a217a165,(xprime,yprime)a52a12a105a24a28a43a41,a183a162a21a25a24a199a47a34,a243a176a145a123a30a127,a135a73a123a198
a27,a183a162a85a196dx ∧ dy,a89a52a24a199a198a27,a14a143a198a17a218dx ∧ dya243a135a232a73a222a52a33a135
a201a123a10a154,a31a143a119a135a73a218a31a143a52dxa218a89a199a52a104a118a234a106a166a27a1a186a35a123a47,a14a143
3
a248a189a135a73a123a189a66a203a196a190a31a143a52dx,a89a199a105a102a38,a44a49a41a116a105a119a63,a88a44a178a199
a89a56a249a137a7a139a24a189a123a30a45,a196a49a183a106a106a141a141a128a139a24a199dx,a243dx,dya89a171a135a73
a131a45a7a79a193a198a27∧,a31a143a119dx ∧ dya218a89a199a175a11a205a133a236a234,a28a152a42dx,dya253a252
a52a31a143a209a88a56a249,a198a234a49a128a52a31a143a192a220a205a52a24a199a105a135a201a104a10a123a175a11,a243a89a48
a193a139a24a199a76a123a159a90,a210a52a90a159a105a83a106a90a105a135a73,a34a189dx ∧ dya89a199a198a27a52a39a233
a193,
dx∧dy =?dy ∧dx,(1.5)
a89a199a175a11a210a56a249a59a92a234,a79a143a198a27a152a42a52a39a233a193a123a155,a104a108dx ∧ dx = 0,a47
a34a222,a79a143dx ∧ dx =?dx ∧ dx,a196a49dx ∧ dx = 0,a243a39a233a193a123a198a27a131a5,
a178dx ∧ dya23a196a35a106,a79a143a198a27a52a39a233a193a123,dx2 = 0,a196a49a210a150a139a176a39a123a192a220
a234,a89a248a122a116a123a83a106a119a41a105a83a106,a89a199a83a106a105a201a123,a139a24a199a193a47a123a136a88a213a166
a35a106a218a42a143
dx∧dy =?(x,y)?(xprime,yprime)dxprime ∧dyprime,(1.6)
a34a38a183a162a123a135a73a126a123a52a160a135a73,a196a49
dx =?x?xprime dxprime +?x?yprime dyprime,dy =?y?xprime dxprime +?y?yprime dyprime,(1.7)
a25a243a126a105a198a27a24a198,dxprime ∧ dxprime = dyprime ∧ dyprime = 0,a13dxprime ∧ dyprimea79a143a198a27a52a39a233a193
a123,a196a49a52a166a80a198a49x = x(xprime,yprime),y = y(xprime,yprime) a123a196a44a7?(x,y)?(xprime,yprime),a89a199a110a82a52
a196a44a7,a52a143a199a160a135a73a196a196a123a113a239a42,a196a49
dx∧dy =?(x,y)?(xprime,yprime)dxprime ∧dyprime,(1.8)
a89a199a166a28a52a183a162a173a232a73a166a35a106a123a24a199a7a248.a183a162a127a119a173a232a73a7a52a166a35a106a123
a155,a199a97a148a198a222a196a44a7,a196a49a89a199a136a88a210a52,a233a173a232a73a123Integral,a253a232a73a5
a123a42a6,a178a232a73a82a191a171,Integrala52a24a199a135a73a245a49a42,a198a27a52a39a233a193a123,a196a49
a34a38a245a173a232a73a1393a145,4a145a116n a145a123a56a45,a245a173a232a73a123Integrala44a23a196a52a105a83
a106a123a245a49a42,a0a143a166a35a106a210a7a108a233a234,a89a176a62a139a24a154a135a201a123a143a48,a79a143a47
a158,a28a7a121a210a166a35a106a123a218a42a123a30a127,a34a189a196a44a7a52a116a123,a88a108a123a155,a198a222a196
a44a7a123a253a233a138,a38a199a52a116a123,a89a199a52a176a145a1a91a135a201a123a192a220,a210a52a56a45a139a199
4
a53(Orietation),a28a221a123a30a127,a1392a199a35a39a221a123a48a53,a221a123a30a127,a34a38a149a234a48a53
a123a155,a196a44a7a52a139a138,a79a36a183a162a24a199a136a88a52a245a173a232a73a123Integrala97a148a52a24a199a105
a83a106a245a49a42,a52dx,dya123a245a49a42,a198a27a52a39a233a193,a89a248a166a35a106a113a92a44a49a233a123,
a104a108a183a144a41a2342a145a123a190a6,a176a145a52a105a210a23a123,a51a248a123.a105a198a27a52a201a122a105a254,a52
a88a204a139a176a39a123,a196a49a7a118a59a92,a178a48a24a5,a210a520.
4 a105a135a73
a222a193a89a234a89a143a248a24a171a7a248,a4a150a89a7a248a164a205a7a80,a183a162a89a176a127a135a232a73a123a30a127,
a24a199a173a7a123a189a174a52a194a245a189a174(Green’s Theorem),a210a52a128,a34a38a28a139a199a75a173,
a243a30a142a222a123a135a73a52a44a49a35a143a75a173a222a123a135a73,a52a24a199a24a173a232a73a90a19a173a232a73a123
a7a248,a89a52a199a58a158a173a7a123a7a248,a7a48a215a12a115a71a139a24a253a66a86,a89a116a89a199a7a248,a198
a128a143a89a52a114a199a135a232a73a123a228a253a189a174,a183a52a51a63a123,a89a248a123a7a248a25a243a47a158a85a194a245
a189a174a123a30a127,a128a128a52a85a196a139a232a73,
integraldisplay
γ
Adx + bdy = (?B?xA?y )dxdy,(1.9)
a152a42a139a24a199a175a11,a139a30a127a28a44a49a144a11Integral,a88a7a11a217a199,a0a143Integrala210a52
a178a24a199a24a39a135a73a42a35a143a220a39a135a73a42,a89a14a143a35a17a218a218a42a189a174a52a89a248a6,a183a210
a90a156a24a199a105a135a73,a183a162a166a98a89dx ∧ dy a52a24a199a245a49a42,a52a24a199a105a83a106a123a24a199
a42a6,a210a54a183a162a202a47a245a49a42a24a248,a88a98a152a36,a233a149a89a248a123a42a6,a183a162a164a44a49a189
a66a199a24a199a135a73,
d(Adx + Bdy) = dA∧dx + dB ∧dy = Aydy ∧dx + Bxdx∧dy,(1.10)
a119a105a135a73(Exterior differential calculus),a105a135a73a105a59a92,a34a247a139Adx + Bdy,
a199a123a135a73a210a52a135a73a199a123a248a106,a15a210a52a135a73a60a106,Aa166Ba52x,ya123a60a106,a196a49
a210a135a73A,B,Aa123a135a73a210a52Axdx + Aydy,Ba123a135a73a210a52Bxdx + Bydy,a44
a52Axdx∧dx = 0 a210a122a116Aydy ∧dx,a145a19a49a210a122Bxdx∧dy,a98a52a79a143a198a27a52
a39a233a193a123,a196a49a210a122(Bx?Ay),a89a52a194a245a189a174a176a622a173a232a73a123a248a106,a196a49a194a245
a189a174a178a92a39a232a73a35a196a220a39a232a73,a199a123Integrala34a19a222a52a199a105a135a73,a44a49a23a241a105
5
a135a73a52a105a201a123a192a220,a79a36a28a44a49a178a232a73a82a191a171,a210a128a183a162a252dx,dya5a24a199a105
a83a106,a233a89a199a105a83a106a139a199a105a135a73,a105a135a73a105a59a92,a210a52a34a38a135a73a200a49a123a30a127,
a217a34a52a233a154a49a248a106a135a73,a136a42a183a122a116a24a199a245a49a42,a89a199a245a49a42a123a39a106a176a24
a199,a42a143a60a106a210a35a143a24a39a135a73a42a234,a196a49a39a106a176a24a199,a79a36a210a42a143a198a117a52ka39
a123a155,a122a116a24a199k + 1a39a123a135a73a42,a89a199a52a194a245a189a174a165a152a91a178a77a34a135a73a123a135a73
a42a35a143a75a173a135a73a42,a24a173a135a73a35a143a19a173a135a73a123a218a42,a89a199a210a105a80a234,a79a143a89
a176a193a139a24a199a105a83a106,a196a49a178a89a199a135a73a42a198a229a117,a126a24a199a105a198a27,a135a73a123a198a27
a52a39a233a193,a108a128a17,a25a243a183a139a24a199a135a73,a199a178ka39a123a105a135a73a42a35a143k + 1a39a123
a105a135a73a42,a89a248a6a210a178a89a199a105a135a73a42a165a45a201a234a24a199a99a123a136a232,a44a49a135a73,a89
a199a135a73a203a202a47a123a135a73a88a24a248,a199a52a178ka39a35a143k + 1a39,a135a73a24a196a143a15a52a28a24
a39,a89a199a105a135a73a52a33a0a30a127Frobenius,Daubouxa90a183a123a144a19Elie Cartana90a159
a117a123,a198a162a33a240a90a159a89a199a10a39a52a233a149a24a39a135a73a42,a52Frobenius,Daubouxa90
a156a123a24a39a135a73a42,a13Elie C artana52a27a41a123a115a71,a52a183a123a144a19,a198a57a89a52a19
a27a45a22,a15a210a52a222a199a45a22a33a149a76a123a1a91a166a27,a27a41a174a169a76a166a123a115a71,a183a46a89
a171a115a71a105a52a220a41,a198a88a41a47a123a217a196,a219a41a106a166,a30a158a213a49a52a113a92a99a123,a139a24
a35,a198a201a234a24a160a49,a52a21a137a219a197a166a22(Analytical Mechanics),a198a178a105a135a73a123
a10a39a44Frobenius,Daubouxa44a24a39a42a123a189a66a77a18a116a176a39a42,a196a49a114a199a123a105a135
a73a52Elie Cartana90a159a117a123,a89a52a139a126a123a192a220,a89a199a105a135a73a139a219a5a123a25a54,a52
a126a220a39a131a128a127a1490.
d2 = 0,(1.11)
a253a89a199a105a135a73a126a220a39a127a1490,a183a162a7a121a210(1.11),a210a52a233a195a88a24a199ka39a135a73a42,
a135a73a24a39a210a35a143k + 1a39,a220a39a210a35a143k + 2a39a135a73a42,a199a24a189a520,a7a121a210a89
a24a154,a183a121a210a233a149a60a106a233a234,a210a113a234,a196a49a183a7a121a210a233a149a127a63a123a60a106f,a178
a89a199d,a105a135a73a126a220a39,a210a127a1490,a253d2f = 0 a210a113a234,a0a143a143a31a143a17a218a79a143
a23a108a183a7a121a210d2 = 0,a144a7a121a210d2a42a126a243a144a139a24a49a222a233a210a113a234,a89a52a79a143a199
a52a34a117a123,a196a49a152a42a34a117a24a49a139a89a199a117a159,a0a143a114a199a123a90a210a127a1490,a0a143a24
a49a123a155,a209a52a24a199a60a106a198a222a24a28dx,a183a25a243a160dxi,a210a52a34a189a243a176a145,a243na145,
xia210a52x1a116xn,a243a176a145a30,a152a42a139a24a199a60a106f,fa52x1,···,xna123a24a199a60a106,a233a149
a89a199a60a106,a126a105a135a73a220a39,a24a189a127a1490,a47a34a222,a79a143a105a135a73a24a39a210a122a116aia52fi
6
a233xia123a24a199a160a135a73,a0a143a242a126a24a39a17,a199a123a248a106a210a52a44xi a116xja135a73ai,ai
a52fa123a233xi a123a135a73,a196a49a89a52fa233a44xia116xja123a19a126a135a73:
d(aidxi) =?ai?x
j
dxj ∧dxi,(1.12)
a89a199a60a106a233a149i,ja52a233a193a123,a47a34a222a183a162a127a119a24a199a60a106a135a73a220a39a123a155a203a39
a147a150a139a7a248,a52a233a193a123,a152a42a24a199a233a193a123a60a106a52dx ∧ dya123a248a106,a13dx ∧ dya52
a39a233a193a123,a0a143a199a210a127a1490 a234,d a52a24a199a105a135a73,a52a233a105a83a106a123a245a49a42a123a24
a199a228a174,a89a199a228a174a228a126a220a39a210a127a1490 a234,a89a52a24a199a234a88a122a123a7a248,a79a143a1a91
a222a89,a34a38a28a139a24a199a75a173,a28a82a89a199a75a173a123a30a142,a242a82a89a199a30a142a123a30a142,a210
a150a139a30a142a234,a34a38a28a82a123a30a142a52a114a199a69,a0a143a69a150a139a30a142,a196a49a1a91a222a89a139
a24a199a228a174a70a30a142,a70a30a142a123a155,a126a220a39,a210a127a1490,a139a24a199a75a173a123a70a24a39a30a142
a52a24a199a105a80a123a75a173,a253a88a242a139a30a142a234,a89a199a1a91a123a117a159a203a105a135a73a123a117a159a52a233
a83a123,a70a220a39a30a142a24a189a127a1490,a89a52a199a1a91a123a117a159; a70a105a135a73a220a39a127a1490,a52
a199a73a219a123a117a159,a89a220a199a192a220a88a52a220a199a144a88a35a7a123a192a220,a52a113a92a233a83a123,a52a24
a195a47,a24a199a30a142a47a158a126a110a82?a44a43,a30a142a220a39a127a1490,a253?2 = 0,a199a203a105a135a73
a52a233a83a123,a89a52a24a199a234a88a122a123a1a91a7a248,a234a88a122a123a106a166a222a123a7a248,a201a122a88a122
a234,a79a143a70a30a142a52a24a199a1a91a123a175a11,a205a52a24a199a114a14a123a175a11,a24a189a7a252a114a199a75
a173a198a222a30a142,a98a52a70a105a135a73a52a199a73a219a123a175a11,a52a199a219a92a123a175a11,a7a105a135a73a144
a7a127a119a89a199a135a73a42a243a24a154a142a163a123a117a159a210a139a234,a89a24a199a219a92a123a228a174a203a24a199a114
a14a123a228a174a139a89a248a233a83a123a7a248a52a105a10a122a123a47a60,a52a24a199a173a7a123a1a91a25a54,a52a173
a7a123a106a166a25a54,a143a31a143a233a83a17a218a217a34a89a210a52a194a245a189a174a123a77a18,a210a52Stokesa189
a174,Stokesa189a174a89,a34a38a139a24a199a75a173,a178a199a85a20a222,?ka52a89a248a24a199ka145a123a75
a173,a196a49a199a123a30a142a210a52a30a142k,a0a143a34a38a139a24a199a135a73a42a119a41α,a199a123a39a106
a52k? 1(degα = k? 1),a149a52a183a162a210a139a89a143a24a199a7a248,α a243a30a142a123a232a73a127
a149dα a243?ka123a232a73,integraldisplay
k
α =
integraldisplay
k
dα,(1.13)
a89a52a173a7a244a234a123a189a174,a47a158a126Stokes a214a66,Stokes a52a93a41a123a97a126a106a166a27,a28
a162a76a150a243a89a199a49a165a46a178a38a116Stokes a189a174,Stokes a189a174a210a178a220a199a202a47a123a228a174,
7
a24a199a52a127a149a75a173a123a30a142a123a228a174,a24a199a52a127a149a105a135a73a123a232a73,a89a220a199a139a59a92a123
a7a248,a34a38a183a162a178a105a135a73a123a232a73a85a196a89a199a7a248,
(,α) = (?,dα),(1.14)
a89a199a105a135a73a196a24a199a37a222a56a45(Vector Space),a44a49a28a62,a89a199a75a173a15a52a12a105a24
a199a37a222a56a45,a15a44a49a28a62,a34a38a89a220a199a37a222a56a45a178a44a232a73,a79a36a210a139a24a199a196
a162a123“a233”(pair),a89a199a37a222a56a45a123a24a154a90a0a199a37a222a56a45a24a154a203a243a24a229a52a122a116
a24a199a116a106,a122a116a24a199a106,a0a143St okesa189a174a210a52a128a89a199paringa38a122a233?a123a42a126
a123a174a6? a166a105a135a73d a52a202a177a123(adjoint),a52a233a83a123“a233”,a89a210a52Stokesa189a174a123
a63a66,a176a145a30,a249a127a63a145a30a209a52a233a123.a215a12a115a71a243a198a123a66a86a176a128,a89a199a52a135a232
a73a123a228a253a189a174,a44a199a210a201a241a183a162a202a47a135a232a73a123a228a253a189a174,a79a143a34a38k = 1,
a0a143a183a162a123a75a173a52a24a199a34a227,a44a a116ba123a34a227,a89a199a34a227a210a52?,a199a123a30a142a17,
a52ba154a62aa154,α a243a89a176a52a24a199a60a106,a222a39a89a123dαa52a199a232a73,a243a24a145a123a60a111a210
a52a126a116a134a34a222,a79a36a243a24a145a123a60a111?a52a199a34a227,a199a123a30a142a52b? a,αa52a24a199a60
a106f,a196a49dαa52df,a149a52
(b?a,f) = (?,df) =? f(b)?f(a) =
integraldisplay b
a
df,(1.15)
a89a210a52a128a60a106a243ba154a123a138a62a86a60a106a243aa154a123a138a127a149dfa243a89a199a34a227a222a123a232a73,
a89a199a210a52a196a162a135a232a73a123a228a253a189a174,a15a210a52a128a141a30a52a44aa116ba232a73df,a38a30a210
a52f(b)? f(a),a89a210a52a183a162a123a228a253a189a174,a196a49Stokesa189a174a52a135a232a73a123a228a253a189
a174a243a176a145a123a77a18,a79a36a243a245a195a123a135a232a73a176a62a15a52a199a159a90,a58a158a139a126,a79a143a105
a135a73a221a57a105a245a97a238,a139a24a199a218a42a105a142a52a121a210a123,a210a52a28a178a220a199a105a135a73a123a42
a6αa203βa35a198,a13a70a89a199a123a105a135a73,
d(α ∧β) = dα ∧β + (?1)degαα ∧dβ,(1.16)
a89a199a218a42a105a142a52a121a210,a79a143a59a92a143a144a7a34a189αa90βa209a92a49a210a113a234,a89a52a132a149a233
a149αa90βa209a52a34a117a123,a34a189a199a162a209a52a92a49a123,a210a44a49a85a196dx1,···,dxk,···,dxn,
a3a62a198a199a60a106a24a174a210a44a49a122a116a234,a196a49a199a162a89a199a198a27a131a45a90a105a135a73a139a89
a248a24a171a59a92a123a7a248,a89a199a7a248a88a98a152a36,a164a44a49a205a207a123,a79a143a34a38a139a24
8
a199a228a174,a199a123a178a48a127a1490,a89a52a105a88a122a234a123,a89a199a210a44a49a5a24a199a248a27,a139a199
a219(quotient),a89a248a122a116a24a199a248a27,a25a243a119a41a51a174(homology),a25a243a142a245a106a166
a123a21a48a209a52a139a199a228a174,a28a220a39a127a1490,a28a210a21a5a24a199quotien t,a14a143a248a17,a31
a143a119quotienta17? a210a52a28a178a196a139a123a119a22dα = 0a123α,a250a196a139dβa117a248,a253
{α|dα = 0}/dβ,(1.17)
a7a52α = dβa123a155,a79a143d2 = 0,a196a49dα = 0,a79a36a28a82a196a139a123a196a162a123a20a111
a42(close form),a250a44a49a85a196da31a143a123a192a220a117a248,a210a122a116a243a106a166a176a62a126a24a199a142
a124a123a214a9a119homology,a15a210a52a82a196a139a123ka39a123a135a73a42,a199a162a52a85a20a123(a250da42
a126a1430),a250a196a139a123dβa123a248,a5a24a199a219a136a232,a89a199a219a136a232a210a119a41homology,a28a44
a49a126a116a89a199d,a15a44a49a126a116a89a199a30a142,a126a116a30a142a123,a187a36a222,a52a243a95a192a176a62,a11
a139a126a30a142a123,a79a143a126a123a52?a123homology a119a222a51a174(cohomology),a89a52a132a149a187
a36a123a7a248,a214a9a126a171a234,a196a49a119cohomology,a89a199a105a184a51,a34a38a28a139a24a199a22
a111,a199a52a155a151a123,a199a123cohomology forma52a139a33a145a123,a89a199a139a33a145a123a145a106a119a89
a199a56a45a123Betti a106(Bett i Number),a89a52a95a192a123a19a142,a92a166a135a232a73,a44a49a88a23
a86a11,a88a44a89a199a11a173a114a199a123a139a173a7a123a21a48,a52a163a117a106a166a123a21a48a228a253a19a142,a104
a108a105a7a155a234,a28a139a24a199a105a76a123a56a45,a196a139a135a73a42a28a196a123a56a45a76a122a88a122a234,a199
a139a136a232,a28a44a49a28a62,a15a44a49a70a105a135a73,a76a122a88a122a234,a108a128a17,a199a139a74a1a91a123
a117a159,a82quotient,a89a199quotienta52a139a33a123,a89a199a139a33a139a199a80a255,a122a116a106a248a139a33,
a52a128a139a33a145a123a145a106a52a245a232,a122a116a24a28a106,a89a28a106a248a210a52a89a199a56a45a123a173a7a117
a159,a79a143a122a127Betti a106a52a24a199a114a106,a139a24a107a114a106a105a7a155,a7a48a128,a69a193,a69a193a139
a89a171Bet ti a106,a162a193a15a139Betti a106,a199a162a52a88a24a248,a5a193a180a95a192a123a124a46a27a7a121
a210a89a171Betti a106a52a95a192a88a35a222,a79a36a95a192a243a106a166a123a228a126a165a210a7a155a234.
9
a145a19a89 a141a106a166a233a106a60a106
2001a3510a21919a134
1 a253a49a123a15a153a90a248a123
a164a139a1a73a168,a183a46a190a89a199a229a204a89a24a89a183a123a15a153a90a248a123,a183a89a199a49a123a49a30
a528a199a66a30,a98a135a232a73a76a122a88a122a234,a135a232a73a123a41a140a105a18,a88a7a1288a199a66a30,a210
a5280a199a66a30a15a89a88a113a123,a196a49a183a104a108a144a21a89a199a76a150,a131a217a52a144a235a114a199a123a139
a24a74a63a66a123a175a11,a150a149a45a251a123a60a111a183a150a27a86a245a89,a88a45a251a123a189a66a221a86a121a210,
a183a46a28a162a46a178a166a44a135a232a73,a196a49a183a209a88a24a189a7a201a28a162a107a31a86,a28a162a195a86a23
a24a23a7a3a49a3a126a123a86,a76a150a243a86a176a73a122a116,a15a139a183a123a89a123a41a140a90a19a142a52a86
a165a150a139a123,a183a250a122a97a148a10a24a10a135a232a73a114a199a123a107a47a221a86a52a243a0a74a48a193a53a3a21
a48,a44a49a128,a135a232a73a53a3a21a48a76a150a139a220a199a33a173a7a123a48a193,a24a199a52a243a1a91a123a97
a126,a135a232a73a243a135a73a1a91a123a97a126,a33a0a52Gauss,Gaussa15a142a88a52a33a0a123,a97a148
a164a139a47a123a124,a152Euler,Mongea127a124,a88a44,a183a46Gaussa5219a45a22a92a45a142a33a149a76
a123a106a166a27,a106a166a243a0a30a127,a92a45a142a15a210a52a220a78a234,a79a143a89a199a198a79,a121a41a123a106
a166a24319a45a22a52a92a45a142a33a80a123,a0a30,a88a98a139Gauss,a164a139Gaussa123a107a47a249a217a166
a9,Gaussa33a7a155a123a166a9a210a52Riemann,a79a143a139Gaussa90Riemann,a121a41a123a106
a166a210a11a11,a11a11a123a63a135a210a52a76a27a203a248a198a123a48a53a86a21a48,a243a1a91a222a123a97a126a123a21
a48a52a105a245a123,a104a35Einsteina17a128a44a212a174a25a54a210a52a1a91a25a54,a49a36a21a48a198a123a18
a66a35a233a88,a18a66a35a233a88a104a108a7a126a43a41,Einsteina234a137a33a240a123a43a41a44a43a1a91a175
a11,a230a130a43a41(x,y) a139a1a91a123a63a66,a104a24a199a212a174a166a27a250a122a97a148a139a1a91a123a221a212
a174a123a63a66a30,a198a41a229a117a98a7a118a92a174,a88a44,Einsteina122a122a234a137a89a199a41a88a116,a79
a143a56a45a17,a117a122a7a118a133a236,a199a227a142a127a63a43a41,a227a142a43a41a123a127a63a160a10,a79a36a15a227
a142a43a41a35a166,a89a210a52a183a162a25a243a196a119a123a22a111,a22a111a123a150a39a52a56a45a150a39a123a77a18.
a253a117a126a123a52Euclida56a45a221a86a58a78a56a45a127a144a139a1a199a56a45,a25a243a77a18a123a22a111a210a114
a199a77a18a234,a77a18a234a49a128,a114a199a123a56a45a10a39a243a212a174a222a107a47a53a3a21a48a234,a79a36a1
a91a176a62a7a195a85a212a174a25a54a210a137a7a1a91a99a123a150a39,a248a234a22a111a131a105,a164a139a14a145a45a123
1
a10a39,a243a5a193a123a49a165,a183a46a227a135a203a76a27a89a24a89a1a91a48a193a123a21a48,a135a232a73a164a139
a24a199a21a48,a33a7a155a123a52a133a106,a105a219a5a123,a202a47a123a106a248a52a34a106,a0a143a243a34a106a173
a222,x2 + 1 = 0 a210a150a139a137,a243a133a106a173a222,a183a162a88a98a38a199a139a137,a63a37a133a106a139a58a158
a28a201a123a117a159,a139a105a245a25a54a209a250a56a243a133a106a176a62a234,a133a106a166a34a106a24a248,a139a228a174a123
a21a70,a28a126a89a199a21a70a131a128,a133a106a83a44a234a105a245a25a54,a183a162a49a128a204a23a116a243a133a106a176
a62a123a89a74a19a142,a196a49,a106a166a7a97a126,a183a162a89a199a49a52a97a126a106a166,a7a166a204a97a126,a7
a97a126a123a155,a204a21a25a133a106a105a7a155,a79a36,a133a35a60a106a88a24319a45a22a123a21a48a52a106a166a176
a62a33a7a155a123,a52a24a199a7a217a199a48a193a123a21a48a117a122a205a7a155a24a74a123a21a48,a33a128,a183a122
a19a154a30a45a89a89a243a133a106a48a193a123a97a126,a133a106a88a144a52a38a122a233a149a127a24a199a48a199a42a139a137,
a63a37a188a126a133a106,a105a245a106a166a175a11a117a122a59a92,a133a35a60a106a88a7a34a35a60a106a88a59a92a245
a234,a34a35a60a106a88a139a142a245a228a54a123a175a11,a217a34a166a34a19a88a76a139a7a248,a88a44a104a30a15a137
a7a234,a196a49a89a52a220a199a11a248,a183a7a243a89a199a49a199a176a62a178a199a162a46a205a27a89a24a154,a38a122
a76a27a21a234a137a135a232a73a243a199a162a222a123a97a126a52a33a173a7a123a220a199a48a53.
2 a7a149Stokesa218a42a123a86a223
a222a39a89a116a234a135a232a73a123a228a253a189a174,a139a30a127a210a85a196a89a171a111a231a213
integraldisplay b
a
f(x)dx =
integraldisplay
f(x)|ba → L(
integraldisplay
f(x)dx) = f(x),(2.1)
a253a89a220a199a42a6a35a127,a105a110a99a143,a104a35a183a243a8a13a135a205a236a39a135a232a73,a183a7a3a210
a139a24a199a175a11,a143a31a143a89a210a52a228a253a189a174,a41a170a88a195,a105a88a115a143,a28a162a76a150a25a243a15
a164a139a89a199a243a14,a88a163a175a175a11,a183a0a30a15a88a163a175a175a11,a203a28a162a25a243a24a248,a41a170
a88a195,a44a234a105a245a35,a98a127a119(2.1)a123a104a52a228a253a189a174,a89a52a79a143(2.1)a128a210a234a135a73
a166a232a73a123a7a248,a89a199a42a6a123a220a224,a24a30a52a199a189a232a73,a52a24a199a193a232,a141a30a52a135
a73a35a39a123a228a174,a196a49a141a30a123a232a73a52a24a199a88a189a232a73,a166a233a155,a52a24a199a60a106,a199a123
a135a73a52f(x),a15a210a52a128,a199a123a38a30a52a232a73,a141a30a52a135a73,a0a143a89a199a228a253a189a174
a210a128a210a234a135a73a166a232a73a123a228a253a7a248,a76a151a222a128,a135a73a52a232a73a123a24a199a39a228a174,a210
a52a7a73a24a199a60a106,a38a122a199a42a143a46a127a60a106a123a135a73,a25a243a123a175a11a52,a116a234a176a145a14
a143a248a218a89a199a228a253a189a174a52a24a199a35a106a123,a25a243a34a247a245a35a106,a204a14a143a248a218a89a210a52
2
a245a35a106a123a135a232a73,a139a24a199na145a123a56a45,na145a5a117a210a139a142a245a88a51a123a145,a245a35a106a123
a135a232a73a228a253a10a39a52a199a173a232a73,a243a178a193a222a52a24a199a19a173a232a73,a243a176a145a123a56a45a52a245
a173a232a73,a183a222a39a89a234,a232a73a139a24a199a232a73a123a75a173,a232a73a52a243a24a199a75a173a176a70a232a73,
a108a128a164a139a24a199a174a6,a204a7a255a88a232a31a143a192a220,a89a24a199a60a106a52a31a143a192a220,a183a222a39
a89a89a199a174a6a52a24a199a105a135a73,a105a135a73a210a52dx,dy a89a74a135a73a198a229a117,a88a44a89a199a198
a27a52a39a233a117a123,a39a233a193a201a244a234,a79a143a39a233a193a131a128,a24a199dx a88a21a234a63a243a220a39,
a253(dx)2 = 0,a24a199a7a155a123a175a11a52a31a143a119dx,a89a199a175a11a7a118a133a236,a89a229a117a7a118
a159,a89a199a175a11a15a210a52a31a143a52a24a199a60a106a123a135a73,a183a162a34a189dxa52a104a189a123,a139a63a66
a123,a49dxa143a35a106a5a24a199a245a49a42,a89a199a245a49a42a123a198a27a52a39a233a193,a89a171a39a233a193a198
a27a123a245a49a42a119a105a135a73a42,a105a135a73a42a210a52a141a232a73a123a24a199a233a54,a243a24a199a75a173a176a232
a89a199a105a135a73,a89a15a44a49a23a42a24a171a123a83(pair),a139a24a199a75a173,a242a139a24a199a232a73a90,
a56a243a24a229,a232a73a139a24a199a138,a89a199a138a52a24a199a106,a89a220a199a52a123a92a123a136a42,a139a234a89a199
a245a173a232a73a123a10a39a131a128,a245a35a106a123a135a232a73a228a253a189a174,a210a52a196a162a123Stokesa189a174,St
okesa189a174a52a24a199a1a91a123a25a54a166a24a199a73a219a25a54a201a92a123a136a42,a243a176a145a30a127,a190a152
a243na145a123a56a45,a34a38a63a243a24a199ka145a123a75a173,a199a44a49a52a132a145a123a127a63a75a173,a139a89a248
a24a199ka145a75a173,a190a152a178a193a222a24a199a19a145a75a173,a56a45a165a24a199a77a193a127,a105a210a23a143,a89
a199a132a145a75a173a139a24a199a30a142,a75a173a139a30a142a123a10a39a52a83a106a95a192a24a199a228a253a10a39,a28a7
a207a196a199a123a30a142a7a248,a24a199a253a47a123a207a196a210a90a116a196a162a195a5a196a51a174a107(homology).
a51a174a107a52a83a106a95a192a207a196a56a45a117a159a123a33a228a253a123a24a199a10a39,a25a243a139a24a199ka145a75
a173?,a199a123a30a142a85a196,a12a105a139a24a199(k?1)a145a105a135a73,a105a135a73a42a6a52k?1a39,
a135a73a49a128a143ka39,a196a162Stokesa189a174,a210a52a128a233a149ωa52a24a199k? 1a145a105a135a73a42,a199
a123a105a135a73dω a243?a222a232a73a127a149a178ωa243?a30a142a222a70a232a73,a253
integraldisplay
dω =
integraldisplay
ω,ω a52a105a135a73a42,(2.2)
a89a199a210a52a196a162Stokesa189a174,a89a52a245a35a106a135a232a73a123a24a199a228a253a189a174,a243a215a12a11a9
a85a123a86a165,a15a1a47a10a241a89a199a10a154,a89a199a228a253a189a174a123a104a221a105a183a222a193a89a123a0a199a228
a253a189a174a42a143a1a47a60a111,a34a38a89a199a56a45a521a145a123,a2431a145a123a60a111,a75a173a52a34a227,a199
a123a30a142a210a52a34a227a123a220a199a224a154,a220a199a154,a70ωa243a220a199a224a154a123a138(a217a165ωa143a0a199a88
a189a232a73)a210a52a183a243a222a193a228a253a189a174a123a218a42a123a141a30a60a106a243ba123a138a62a86a243aa123a138,a210
3
a52a243a30a142a222ωa123a232a73,a13a38a30a210a52a243a89a199a34a227a123a232a73,a210a52a44aa116ba123a189a232a73.
a196a49,a88a10a23a241a117Stokesa189a174a243a134a34(1a145)a123a60a111a210a52a135a232a73a123a228a253a189a174,a0
a1432a145a123a60a111a17a2182a145a210a52a105a139a214a123a196a162Greena189a174:
integraldisplay
(Px?Qy)dxdy =
integraldisplay
Pdx + Qdy (2.3)
2a145a60a111a30,a75a173a522a145a123,a199a123a30a142a52a77a34,a149a52(2.3)a210a52a178a158a123Greena189
a174,a28a162a209a127a119,a209a105a89a231,a196a49Stokesa189a174a243a178a193a222a123a1a47a60a111a210
a52Greena189a174,Stokesa189a174a139a88a51a123a214a9,a23a28a126a253a24a253a86,a88a44a25a243a7
a118a47a113a119Stokesa189a174,a0a143a164a139a12a105a123a24a199a1a47a60a111,a243a174a145a56a45,a34a247a139
a24a199a77a193,a199a52a24a199a174a145a75a173a123a30a142,a0a143CIa36a30Storkesa189a174a210a85a196
integraldisplay
(Px + Qy + Rz)dxdydz =
integraldisplay
Pdydz + Qdxdz + Rdxdy,(2.4)
a28a162a243a166a176a127a135a232a73a46a178a141a116a234,a24a199a19a39a42a243a30a142(a77a193)a222a123a173a232a73a127a149
a199a123a174a39a42a243a75a173a176a62a123a174a173a232a73,a89a52Stokesa218a42a123a12a105a24a199a60a89,a114a199
a123a60a89a243a176a145a209a233,a139a24a199a228a253a117a159,a210a52a105a135a73d a126a220a39a24a189a127a1490,a34
a38?megaa52a24a199a105a135a73a42,a0a143
d(d(ω)) = 0,(2.5)
a89a199a48a199a58a158a142a52a121a210,a233a149ω,a105a135a73a42a23a108a52a34a117a123,a196a49a28a144a137a7
a178ωa104a196a24a199a92a49a117a121a210a210a113a234,a89a52a79a143a28a154a24a49a123d2a209a127a1490,a149a52a233a149
a92a49a123a60a89,a92a49a52a24a28da198a222a24a199a60a106,a23a108,a144a7a121a210a24a199a60a106a126a220a39d,
a199a24a189a127a1490a210a44a49a234,a183a142a5a174a234a24a5,a243a24a199na145a123a56a45a165,a199a123a43a41
a52(x1,···,xn),a139a24a199a60a106a52f a52xia123a60a106,da24a39a123a155,a210a52a202a47a123a160a135a73,
a15a210a52?f?xidxi,a242a135a73a24a39,a122a116a19a255a160a135a73,a242a198dxj ∧dxi,a89a199a19a126a160a135
a73fija52a233a193a123,a89a52a79a143a70a160a135a73a166a39a147a195a7,a79a36a89a199a248a106a52a233a193a123,a13
a183a162a89a220a199dxi,dxja123a198a27a52a39a233a193a123,a23a108a220a39a135a73a131a128a210a127a1490a234,a253
d(df) = d(fidxi) = fijdxj ∧dxi = 0,(2.6)
a89a176,a79a143a251a189a234ia166j,a210a122a116dfij?dfji,a98a52a79a143fa52a233a149a89a199a141a41a52a233a193
a123,a196a49a210a520a234,a79a36a222a193a121a210a234a233a149a60a106a123d2 = 0,a89a210a44a49a234,Stokesa189
4
a174a44a49a128a75a173a166a105a135a73a52a24a199a233a83,a38a122a70a30a142a203a174a89a199da89a220a199a174a6a52adj
oint,a52a233a83a123a174a6,a89a52a199a234a88a122a123a136a42,a79a143a70a30a142,a52a24a199a1a91a228a174,
a217a34a70a24a199a75a173a123a30a142a52a24a199a113a92a123a1a91a123a228a174,a52a114a199a123a75a173a123a24a199a117
a159,a70a105a135a73da52a24a199a219a92a123,a73a219a123a228a174,a52a113a92a219a92a123,a144a166a89a24a154a123a142
a163a139a7a248,a196a49a24a199a52a114a14a123a1a91a174a6,a24a199a52a219a92a123a73a219a174a6,a199a162a52
a233a83a123,Stokesa189a174a128a199a162a52a233a83a123,a196a49a89a52a24a199a173a7a244a234a123a189a174,a183a44
a49a5a193a227a135a89a122a245a24a154,a56a45a88a24a189a52a202a47a123Euclida56a45,a15a142a56a45a252xa41
a43a41a143a196a162a123a22a111,a34a247a56a45a52a24a199a22a111a123a155,a15a44a49a255a88a199a123a105a135a73a42,
a190a152ka39a123a105a135a73a42,a24a199ka39a123a105a135a73a42a28a12a105a24a199ka39a105a135a73a42a164a52a24
a199ka39a123a105a135a73a42,a149a52a196a139a123ka39a123a105a135a73a42a196a143a24a199a183a162a196a162a123a37a222a56
a45(vector space),a243a217a165a44a49a159a113a28a62,a25a243a183a210a255a88a196a139d = 0a123a89a171a105
a135a73a42,a253a105a135a73a1430a123a0a74a105a135a73a42,a243a106a166a222,a183a162a193a89a171a105a135a73a42a52a85
a20(close)a123,a89a74a85a20a123a105a135a73a42a232a196a37a222a56a45,a79a143a220a199closea105a135a73a35a28
a133a143a85a20a123,a247
Γk = {ω|ω a52k a39a105a135a73a42},Ck = {ω|ω ∈ Γk,dω = 0},(2.7)
a0a143a183a82ω,ωa52a24a199a85a20a123a105a135a73a42,a25a243a183a178Cka104a196a24a199a107,a89a199a107a139a199
a6a107,a89a199a6a107a52a31a143a17a218a199a210a52a196a139a123k? 1a145a123a105a135a73a42a6a126da117a42a126.
a79a143d2 = 0,a196a49a199a210a24a189a52closea123,a79a36a243a196a139a123a85a20a123ka39a105a135a73a42a232
a196a123Ck a165,a196a139da198a222a24a199k?1a39a105a135a73a42a196a143a24a199a6a107,a149a52a114a199a107a126a6
a107a24a248,a243a107a88a176a62a128a199a52a24a199a219a107(quotient),a89a199a107a139a199a214a9a119de Rham
group:
Hk = Ck/dΓk,(2.8)
a89a243a95a192a222a58a158a173a7,a210a52a128,a105a135a73a42a245a122a88a122a234,a4a150a149closea123a105a135a73
a42a15a245a122a88a122a234,a13a243a28a248dβa131a128,a243a105a245a60a111a131a5,a210a35a196a24a199a139a33a145a123
a37a222a56a45,a0a143a89a199a139a33a145a56a45a123a145a106a52a89a56a45a123a24a199a173a7a123a117a159,a47a158a119
a41Betti number,a89a52a83a106a95a192a165a33a6a123a24a199a228a253a10a39,a15a210a52a183a162a255a88a105
a135a73a42a44a49a251a189a199a123a139a74a95a192a123a88a35a42.
5
3 a141a106a90a233a106a60a106
a183a25a243a86a89a12a105a24a199a175a11,a222a39a139a124a89,a233a149a203a89a199a49a139a74a104a10,a183a89a123a89
a74a11a248a88a24a189a139a7a248,a196a49a28a152a42a233a236a24a199a11a248a139a104a10a123a155,a210a38a183a89a24a199
a47a123a11a248a234,a196a49a88a24a189a73a245a232a107a47,a25a243a183a166a199a11a248,a135a232a73a17a108a52a207
a196a60a106a123a117a159,a126a135a232a73a117a44a43a199a123a117a159,a0a143a60a106a52a245a122a88a122a234a123,a60a106
a139a171a171a123a117a159,a13a139a24a74a60a106,a7a118a59a92,a79a36a15a7a118a173a7a63a37a142a245a97a126a222
a15a141a116,a139a220a199a1a47a173a7a123a60a106a52a141a106a60a106(exponential function)a166a233a106a60
a106(logarithm fun ction),a89a220a199a60a106a139a31a143a117a159a17? a89a52a58a158a173a7a123a60a106,
a183a162a209a65a122a62a24a199a123a135a73a42,a13xn+1a123a135a73a52a127a149(n + 1)xn,a79a36xna123a232a73
a52a127a149 1n+1xn+1,a89a176a34a38n + 1 negationslash= 0,a253
d
dxx
n+1 = (n + 1)xn →
integraldisplay
xndx = 1n + 1xn+1,n + 1 negationslash= 0,(2.9)
a152a42na88a127a149?1,a202a47a124a116a89a199a30a127a210a136a101a234,a79a143a28a127a119a89a199a218a42a52a31a143
a30a127a196a193,a89a199a218a42a243n =?1a30a88a233,a89a210a234a234,a89a52a105a7a108a123,a88a44,a152a42
a89a30a127a7a42a142a123a155,a28a210a150a139a126a116a60a106a232a73a123a173a7a123a189a174,a79a143n =?1a30,
a89a199a232a73a98a139a63a135,a196a49,a34a38n =?1,a183a210a82a233dx/xa123a232a73,a79a143a183a88
a82x = 0,a196a49a183a89a199a232a73a34a189a199a441a232a116x,a89a199a232a73a52a7a155a244a234,a139a63a66a244
a234,a79a143a183a89a199a232a73,a119a199logx:
integraldisplay x
1
dx
x = logx,(2.10)
a89a210a52a233a106a60a106,a5a193a183a255a88a233a106a60a106a123a33a173a7a123a117a159,a34a38a183a178xa198a158
a106a,a233logaxa70a135a73,a132a149loga123a135a73a127a1491/x,a149a52logaxa15a52a127a1491/x,a196a49
a89a220a199a60a106a139a24a199a158a106C:
d
dx log(ax) =
1
axa =
1
x =? logax = logx + C,(2.11)
a34a38a183a82x = 1 a83a156(2.11),a36a30log1 = 0,a89a52a79a143a232a73a52a441a116x,a196a49
a441a1161a232a73a104a108a520,a149a52a183a210a121a116a158a106Ca210a52loga,a79a36a210a122a116loga89a199a60
a106a123a228a253a117a159:loga60a106a126a116axa123a155a127a149logx + loga:
log(ax) = loga + logx,(2.12)
6
a166a233a155a128,a233a106a52a38a122a198a27a35a143a28a27,a89a52a44a3a126a233a106a44a15a174a123a24a199a228a253
a117a159,a25a243a79a143a139a15a174a229a234,a76a150a88a76a126a234,a88a44loga89a199a60a106a58a158a7a155,a79
a143a126a116a234a183a162a89a199a228a253a123a117a159(2.12),a0a143a132a89a199a233a106a123a60a106a193a47a210a90a159
a141a106a123a60a106,a141a106a60a106a52a233a106a60a106a123a39a60a106,a34a38y = logxa123a155,a210a149a189a66,
x = ey,a253
y = logx? x = ey,(2.13)
a79a36a199a162a144a35a52a35a39a123a60a106,eya52a199a141a106a60a106,a217a166logxa24a229a139a28a27a166a198a27
a7a248a123a218a42,logxa178a198a27a35a143a28a27,a141a106a60a106a15a210a178a28a27a35a143a198a27a234,a24a199
a178a198a27a35a143a28a27,a24a199a112a234a44a117,a199a210a178a28a27a35a143a198a27,a89a52a24a199a59a92a123a218
a42:
ex+y = exey,(2.14)
a183a89a176a139a199a121a210:
ex+y = elogu+logv = eloguv = uv,u = ex,v = ey,(2.15)
a89a74a209a52a105a142a52a123a15a174,a183a25a243a7a121a210a141a106a60a106a199a123a135a73a210a52a199a7a3,
a253eya233ya70a135a73a210a127a149ey,a89a199a121a210a143
dy = dxx? expydy = dxdy = x = expy,(2.16)
a89a176a178a141a106a60a106a85a196expy,a104a108a15a44a85a196ey,a89a30a34a189a196a139a123a106a209a52a116
a123,a196a49a150a139a31a143loga63a243a166a100a123a175a11,a149a52,a141a106a60a106a123a135a73a210a52a199a7a3,
a222a193a210a201a241a234a24a199a121a210,a76a27a15a142a16a122,a220a199a60a106a123a67a52a89a199a248a6,a24a199
a52loga243xa123a75a173a165a243x = 0a123a142a163a214a117a214a66a229a117,a74a149a139a195a65,a233a106a60a106a15a52
a24a199a15a159a123a60a106(increasing funct ion),a88a44a199a15a159a122a58a158a122,a141a106a60a106a210a15
a159a122a105a79,a199a123a207a52a116a123,a89a52a183a152a123a220a199a59a92a123a67(graph).a183a46a28a162a243a127
a91a135a232a73a123a86a209a23a116a44a89a220a199a60a106a123a67,a141a106a60a106a166a233a106a60a106a52a58a24a123a60
a106,a24a199a52a12a105a24a199a123a39a60a106,a89a199a117a159a52a58a158a7a155a123,a139a219a201a123a117a159,a145
a24,a141a106a60a106a123a135a73a52a199a7a3,a79a36,a199a139a24a199a105a59a92a123a195a65a255a106,a89a199a195a65
a255a106a52a126Talyora218a42a48a13a123,a183a178Taylora218a42a85a24a5a213
f(b) = f(a) + f
prime(a)
1! (b?a) +
fprimeprime(a)
2! (b?a)
2 +···+ Rn,(2.17)
7
Rn =
integraldisplay b
a
f(n)(t)(b?t)
n?1
(n?1)! dt,(2.18)
a183a46a89a74a28a162a209a39a44a234a89a74a234,a89a199Taylora218a42a178a127a63a123a60a106a48a196a24a199a195a65
a255a106,aa52a24a154,ba52a12a105a24a154,a0a143a199a44a49a48a196(b?a)a123a24a199a245a49a42,a128a193a139
a24a199a155a49,a89a199a155a49a52a132a232a73(2.18)a201a241,Taylora218a42a243a24a199a155a49a123a30a127a210
a52a89a199a196a162a165a138a189a174a195Mean Value Theorema196,Taylora218a42a210a52a165a138a189a174a123
a176a39a123a24a199a77a18,a132Taylora218a42,a25a243a183a162a89a199a141a106a60a106a59a92a122a88a122a234,a79
a143a135a73a5a86a209a52a199a7a3,a196a49a139a24a199a195a65a255a106,a105a59a92,a183a85a143
ex = 1 + x + 12!x2 +···+ 1n!xn +···,(2.19)
a196a49a89a199a141a106a60a106a139a24a199a105a59a92a123a48a13,a199a139a24a199a173a7a123a117a159,a183a46a139a30a127,
a28a89a199a106a248a104a108a52a116a106,a150a232a52a34a106,a139a123a30a127a28a126a24a5a133a106a123a155,a139a105a28
a201a123a117a159a188a51a248a123,a183a127a119sinxa166cosxa139a12a105a89a220a199a48a13:
sinx = x? 13!x3 + 15!x5?···; (2.20)
cosx = 1? 12!x2 + 14!x4?···,(2.21)
a28a204a21a25,a34a38a233ex,a82xa149a143ix,a217a165i2 =?1,a0a143eixa210a139a199a42a6:
eix = cosx + isinx,(2.22)
a183a46a105a142a52a132(2.19)-(2.21)a23a241a117a139a89a248a123a218a42,a227a142a35a106a82a133a106a123a138,a82
a24a199a192a220,a0a143a34a38a28a126a89a199a218a42a123a155,a82x = pi,a149a52eipi =?1,a28a162a209a127
a119a89a199a218a42,a88a44a89a52a199a105a139a63a135a123a42a6,a79a143a89a176a62a139a1a199a158a106,a76a27
a213a63a123a24a199a52pi,a12a105a24a199a210a52e,ea52a79a143Euler,Eulera243a104a3018a45a22a123a0a199
a30a127,a0a30a203a25a243a30a45a88a212a24a248,a0a199a30a127a45a142a210a52a220a78,a45a142a139a41a166a123a21
a48,a210a52a243a220a78,a76a27a203a128a139a24a199a33a149a76a123a106a166a27,Eulera52a0a30a250a203a128a33a149
a76a123a106a166a27,a196a49a139a124a41a234a41a124a131a128,a243a198a123a170a41a176a209a63a139a199a149a76a123a106a166
a27,a149a52Eulera210a250a62a116a18a10a122a227,a198a210a85a234a105a245a105a245a86,a89a199Eulera52a105a139
a63a135a123,a76a150a85a123a169a57a52a150a139a124a167a44a123,a198a85a234a1a186a253,a198a139a80a245a66a47,a196
8
a49a198a52a230a248a66a47a6,a47a6a43a243a198a79a222,a41a198a123a106a166,a183a203a198a17a178a21a9a24a199a7
a248,a210a52a183a162a89a199a8a13a67a86a12a7a88a7a198a123a92a248,a198a123a92a248a7a1a250a76a155a151,a1
a186a253,a105a230a11a123,a128a117a183a251a189a88a111a234,a212a34a234a63a37a57a89a150a139a124a23a234,a169a57a209
a52a110a182a169a123,a136a42a183a162a67a86a12a150a139Eulera123a92a248,a183a162a139a105a245a217a199a124a123a92a248,
a88a44a25a243a23a117a175a11a52a139a74a169a9a123a175a11,a7a48a128,a28a162a152a42a139a213a101a123a155,a44a49
a23a23a176a132a123a92a248,Gaussa166a98a183a128a5219a45a22a33a149a76a123a106a166a27,a76a27a209a127a119a198
a123a106a166a21a197,a196a49a128a17,a152a42a124a27a44a129a62a198,a81a248a198a24a229a86a44a129,a0a143a44a129a30
a210a175a198a66a123a1a91a175a11,Gaussa104a108a210a21a137,a0a143Gaussa123a66a123a1a91a175a11a123a137a243
a198a123a88a169a248a176,a89a52a105a139a63a135a123,a89a74a175a11a113a92a52a240a127a1a91a123,a139a123a210a24a16.
a41a41a198a123a66a123a1a91a189a174,a198a123a66a189a174a209a105a139a63a135,a183a162a44a49a60a198a123a117a85a24a159
a169a57,a105a88a115a143a52a198a123a169a57a88a52a110a182a169,a210a52a121a169,a150a232a52a121a169,Eulera85a234
a105a245a192a220,a89a199a218a42a52a217a165a131a24,a199a178a1a199a204a7a123a106a203a229a117,e,i,pi a139a89a248
a123a7a248,eipi =?1,a89a52a58a158a139a63a135a123,a183a242a183a170a28a162a12a105a24a199a218a42,a88a127
a28a162a139a195a108a85,a51a248a126a89a199a48a13a123a155,a44a49a126a116arctgx,arctgxa123a135a73a52 11+x2.
a196a49arctga89a199a60a106a139a24a199a117a159,a210a52a24a134a70a199a162a135a73a123a155,a42a6a204a7a118a59a92.
a79a36a132a89a74a210a122pia123a24a199a42a6,a253pia44a49a85a196a24a199a195a65a255a106
pi
4 = 1?
1
3 +
1
5?
1
7 +··· (2.23)
a89a52a24a199a164a224a244a234a123a24a199a218a42,a199a178a24a1991,3,5,7,9a89a143a203a243a24a229,a28a243
a24a229a52pi/4 a188a196a49,a89a52a164a224a244a234a123a24a199a218a42a188a163a83a123a24a199a139a214a123a106a88
a27Selberg,a198a52a73a133a123a106a166a27,a139a24a39a198a85a24a199a169a57,a89a198a233a106a88a21a9a108a85
a210a79a143a198a23a116a89a199a218a42,a198a57a89a25a243a52a181a106a76a234a24a154,a183a46a198a44a49a15a13980a245
a181,a124a164a243.
4 a243a1a91a222a97a126
a164a139a1a73a168,a5a24a39a183a7a89a24a154a1a91,a135a232a73a243a1a91a222a123a97a126,a1a91a222a123a97
a126,a104a108a52a56a45a1a91a33a139a63a135a123a52a77a193a123a1a91a9a9a19a145a77a193a123a1a91,a19a145a77
a193a139a171a171a248a6a199a139a171a171a123a111a42a199a139a171a171a88a51a123a117a159a2a243a89a48a193a139Gaussa123
9
a211a42,a210a52
ds2 = dx2+dy2+dz2 = Edu2+2Fdudv+Gdv2,x = x(u,v),y = y(u,v),z = z(u,v)
(2.23)
a89a199a77a193a47a158a126a107a106a44a43a199a243a89a199a174a145a56a45a165,a43a41a52(x,y,z),a149a52a178
a174a145a56a45a123a43a41a44a43a143a220a199a35a106a123a60a106,a196a49x,y,za52a220a199a35a106u,va123a60
a106a199a183a162a193a131a143a107a106(par ameter).a149a52a89a199a56a45a176a62a139a24a199a225a229a172ds2a199a210
a52dx2 + dy2 + dz2,a178x,y,z a44a43u,va123a60a106a131a128a199ds2 a210a35a143a24a199a19a39a123a135
a73a42,a217a165a248a106,a24a196a210a119a41E,F,G,a183a162a65a122a1a91a13a41a123a30a127a139Euclida1
a91a199Euclida1a91a105a149a76a188a199a52a62a24a253a114a199a123a1a91,a199a23a241a117a7a139a24a171a218a174a122
a116a106a166a123a136a88a52a105a173a7a123a199a79a143a106a166a136a88a52a132a24a199a218a174a178a44a24a199a92a246a123a77
a174a122a117,a79a36,a89a199a52a105a228a14a105a42a251a123a24a199a136a88,a13a37a199a217a34a88a144a52a1a91,
Euclida89a199a21a1a91a198a253a22a52a114a199a123a106a166,a198a23a241a117a132a218a174a126a92a246a48a27a77a241a136
a88a123a173a7a117,a233a149a89a48a193a183a250a122a105a110a99a123a52a199a165a41a150a139,a183a162a89a199a49a52a97a126
a106a166,a233a149a97a126a106a166,a165a41a212a213a173a97a126a234,a127a91a192a220a209a24a189a7a139a97a126,a13a233
a149a89a248a123a24a199a106a166a76a27a49a143a150a139a31a143a97a126,a217a34a33a240a28a7a123a52a33a240a41a234a24
a154a123a155,a97a126a204a117a123,a63a37a97a126a205a173a7,a205a253a47,a233a149a89a199a19a39a135a73a42(2.23),
Gaussa123a211a42a210a52a44a49a202a226a89a199a19a39a135a73a42,a21a48a24a199a1a91,a89a199a1a91a210a76
a122a88a122a234,Euclida1a91a44a49a199a123a135a73a42a210a52du2 + dv2,a89a52a24a199a33a59a92a123a60
a111,a12a105a24a199a60a89a131a5a210a44a49a21a48a58a78a1a91,a25a243E,F,Ga52a127a63a60a106,a242a28
a222a24a74a55a104a30a71,a89a199a1a91a123a10a39a18a76a122a88a122a234,a210a52a128,a243a174a145a56a45a44a49
a139a35a193,a253a35a193a222a123a1a91a199a44a49a88a11a89a199a77a193a243a174a145a56a45a176a123a160a152,a144a31
a68a243a77a193a222a123a89a199a1a91,a199a221a105Euclida1a58Euclida1a91a127a243a19a199a18a122a234a88a122.
a149a52,a89a248a123a24a199a21a25a38a122a1a91a139a105a245a199a78a60a1a91a123a10a39a15a210a77a18a116a24a196
a123a60a89,a196a49,a183a5a39a7a89a154a135a232a73a243a1a91a222a123a97a126,a183a46a152a21a15a142a139a88a245,
a89a113a234,a92a92a188
10
a145a174a89 a77a34a88
2001a3510a21926a134
1 a178a193a77a34
a183a46a89a1a39a203a76a27a89a24a154a135a232a73a243a1a91a222a123a97a126,a89a52a58a158a7a155a123a21a48.
a0a143,a44a33a59a92a123a60a89a13a41,a183a162a210a89a178a193a222a123a77a34,a34a247a178a193a222a139a24a30a77
a34x(t) = (x1(t),x2(t)),a253a243a89a199a67a222a196a243a123a60a89,a126a135a232a73a123a155a17,a210a52a89
a30a77a34a139a30a35a34,a35a34a139a199a35a37a222,a233a149a35a37a222,a183a162a82a89a199a37a222a52a92a160a37
a222,a199a123a159a221a521,a15a210a52a82a143a92a160a35a37a222,a149a52a183a162a127a119a34a38a178a43a41xa44a43
a196a140a159sa123a60a106a123a155,a89a210a44a43a89a199a92a160a35a37a222a210a52x a233s a123a135a73dxds,a253a92a160
a35a37a222a143
e1 = (dx1ds,dx2ds ),(e1,e1) = 1,sa52a140a159,eqno(3.1)
a0a143a14a143a248a207a196a89a30a35a34a17a218a105a59a92,a0a210a52a139a234a24a199a92a160a35a37a222a131a128,a63
a34a247a152a42a178a193a52a189a53a123,a253a139a24a199a221a196a123a48a53,a0a143a199a210a139a24a199a92a160a27a53a222,
a15a210a52a203a199a18a134a123a0a199a92a160a37a222,a25a243,a183a119e1a52a92a160a35a37a222,e2 a52a92a160a27
a37a222,a149a52a7a122a116a89a30a35a34a123a117a159,a145a24a71a47a60a210a52a178e1a89a199a60a106a233a149sa242a70
a135a73,a0a143a242a70a135a73a131a128,a104a108a89a52a24a199a99a123a37a222,a79a143e1a52a24a199a92a160a37a222,
a196a49(e1,e1) = 1,a0a143a178a199a135a73a24a5a6,a183a162a210a122a116de1ds a51e1a18a134,a196a49a199a24
a189a243a27a34a123a48a53,a79a36,a183a162a210a139de1ds a127a149a92a160a27a37a222e2a123a245a106,a89a199a245a106a52
a140a159a123a24a199a60a106,a183a162a119k(s),a89a199a245a106a119a22
de1
ds = ke2,e2 a52a92a160a27a37a222,(e1,e2) = 0,(3.2)
ka89a199a60a106a24a196a119a41a77a71,a52a89a30a77a34a243a89a199a178a193a176a62a33a7a155a123a24a199a117a159,
a52a140a159a123a24a199a60a106.
a243a11:
a233a149a201a189a123a77a34a48a199,a201a241a77a71k a123a218a42,[a10a43a213ka52a77a34a48a199a123a24a126a90
a19a126a135a73a123a24a199a60a106]
1
2 a56a45a77a34
a44a178a193a77a34a242a159a24a90a14a143a248a17a218a183a162a23a23a56a45a123a60a111,a34a247a183a162a25a243a139a24
a30a77a34a52a56a45a123a77a34x(t) = (x1(t),x2(t),x3(t)),a2433a145a56a45a176a139a35a34,a196a49
a89a199a56a45a123a43a41x,y a210a44a43a143a107a106t a123a60a106,a89a74a192a220a28a162a76a150a209a127a119,a183
a242a167a243a24a5a6,a196a49x(t) a52a24a199a37a222,a199a123a73a222a210a52a154a123a43a41,a13a154a52t a123
a60a106,a149a52a199a210a42a24a30a35a34,a893a199a73a222a119a41xi(t)(i = 1,2,3),x(t)a52a24a199a37
a222,a52a107a106ta123a60a106,a199a1233a199a73a222a210a52a178a154a123a43a41a44a43a143t a123a60a106,a0a143a14a143
a159a113a17a218a51a248a123a48a27a210a52a233a89a30a77a34a126a135a232a73,a34a189a77a34a139a35a34,a63a37a243a35
a34a222a193a139a92a160a37a222,a253a139a92a160a35a37a222,a233a77a34a139a199a48a53,a24a134a89a143a18,a215a248a24
a199a48a53,a7a152a128a107a106t a52a15a28a123a24a199a48a53,a15a13a211a131,a183a162a210a139a24a199a92a160a35
a37a222,a119a199e1,a0a143a203a178a193a51a248a123a60a89,a178e1a89a199a60a106a233s a242a70a135a73,a210a127a149
a128a233s a70a19a126a123a135a73,a79a143e1 a52a92a160a37a222,a196a49a122a116a123a89a199a135a73a203e1 a52a18a134
a123,a13a233a149a24a30a56a45a165a123a77a34,a199a123a35a34a52a24a30,a199a123a27a34a139a195a106a199,a217a34
a178a44a89a24a154a203a35a34a18a134a123a27a34a139a195a106a199,a0a143a217a165a139a24a199a52de1ds,a199a52a88a113a92
a104a189a123,a132a149a51a248a123a174a132,a183a178a89a199a48a53a123a92a160a37a222a119a41e2,a0a143a199a210a52ke2,
a253a44a49a85a196 de
1
ds = ke2,(3.3)
e2a52a217a165a24a30a27a34,a183a162a119a199a143a204a27a34(principal normal),a13ka89a199a60a106a166a178
a193a123a60a111a24a248,a119a41a77a34a123a77a71,a196a49a25a243,a183a139a24a199a35a34a123a48a53a90a24a199a204
a27a34a48a53,a243a174a145a56a45a210a139a12a105a24a199a48a53a203a89a220a199a48a53a18a134,a183a162a47a158a119
a199binormal,a15a13a211a131,a63a243a12a105a24a199a27a34,a196a49a199a139a220a199a27a34,a0a143a89a220a199
a27a34a196a196a123a178a193a210a52a27a178a193,a77a34a521a145a123,a35a34a521a145a123,a196a49a27a34a522a145a123,
a52a199a178a193,a0a143a139a24a199e3 = e1 ×e2,a217a165e1 ×e2 a52a37a222a232,a233a149a220a199a48a53a123
a155,a139a24a199a104a189a123a145a174a199a48a53a52a203a199a162a18a134a123,a63a37a38a122e1,e2,e3a196a24a199a116
a98a248a58,a183a162a34a189a89a199a56a45a52a189a53a123,a141a67,a38a67a209a139a24a199a104a189a123a48a53,a47
a158a183a162a209a126a141a67,a196a49a210a139a24a199a104a189a123e3a119a22
(ei,ej) = δij,(3.4)
2
a79a36a243a207a196a174a145a1a91a123a30a127,a89a248a123a174a199a144a35a18a134a123a92a160a37a222a196a196a123a67a111
a58a158a7a155,a89a52a143a31a143a17a218a89a248a24a199a192a220a183a119a199a41a37,a28a178a24a199a41a37a194a116a12
a24a199a41a37a123a228a196a52a113a92a104a189a123,a0a143a174a145a56a45a33a7a155a123a117a159a210a52a174a145a56a45
a123a228a196,a183a162a7a207a196a123a1a91a117a159a52a178a44a228a196a88a35a123,a196a49a210a7a127a119a31a143a30a127
a28a44a49a178a89a199a192a220a194a116a12a105a24a199a160a152,a31a143a30a127a199a123a160a152a35a139a243a149a24a199a228
a196,a13a41a37a210a52a89a199a228a196a137a219a123a44a43a123a48a27,a28a7a21a234a194a44a86a210a44a43a89a199a41
a37a194a116a12a105a24a199a41a37a123a228a196a52a113a92a104a189a123,a23a108,a144a139a24a199a228a196a63a37a24a189
a139a24a199a228a196a178a24a199a41a37a35a143a12a105a24a199a41a37,a79a143a7a207a196a56a45a178a44a228a196a88a35
a123a117a159,a196a49a137a219a123a48a27a210a52a188a126a41a37,a0a143a34a38a183a25a243a139a24a30a77a34,a183a88
a144a139a24a199a41a37,a89a74a41a37a52a30a45a123a60a106,a243a0a176a228a196,a79a36eia89a174a199a42a143a41
a37a123a37a222a209a52a30a45t a123a60a106,a149a52a183a44a49a70a199a123a135a73deidt,deidt a52a199a37a222,a79a143
a52(e1,e2,e3)a52a199a41a37,a196a49a127a91a24a199a37a222a210a44a49a85a196e1,e2,e3a893a199a37a222a123
a34a117a28a92,a0a143a183a178a199a227a135a77a18a24a154,a210a178a199a85a196deia127a149eja123a34a117a28a92:
dei = ωijek,(3.5)
a89a199a28a92a123a248a106a52a24a39a135a73a42,a89a52a79a143a183a25a243a41a234a24a5a135a73,a132a89a60a106a34
a122a116a24a39a135a73a42,a0a143a89a248a123a24a39a135a73a42a183a119a199ωij,a89a210a44a43a220a199a35a249a123
a41a37a123a7a248,a28a139a24a199ei a123a41a37,a107a30a139a199a35a249a123a41a37ei + dei,a0a143dei a44
a143eia123a60a106a123a30a127,a199a123a28a92a123a248a106a210a52a24a39a135a73a42,ωij a52a24a39a135a73a42,a243
a89a176,a183a123a141a41i,ja209a52a441 a1163,3 a52a183a162a56a45a123a145a106,a196a49ωij a24a225a1399a199:
i,j a154a24a199a441a1163,a196a49a24a225a1399a199.a899a199a24a39a135a73a42a52a139a7a248a123,a199a88a52a113
a92a127a63a123,a89a199a7a248a210a52
ωij + ωji = 0,(3.6)
a196a49a34a38a28a178ωij a23a196a24a1993 × 3 a123a48a106a123a195a163a123a155,a89a199a48a106a52a39a233a193a123.
a89a24a28a48a199a105a142a52a44ei,ej a123a19a232a127a149δija122a116,a178a89a199a7a248(a48a199(3.4))a135a73
a123a155,a210a193a47a122a116a89a24a28a117a159,a89a210a52a128(ωij) a52a24a1993 × 3 a48a106a123a195a163,a89
a199a48a106a123a195a163a209a52a24a39a135a73a42,a63a37a89a199a48a106a114a199a52a39a233a193a123,a166a233a155a128,
a89a199a48a106a204a110a34a222a123a195a163a520,a217a199a17,a132a149a39a233a193,a139ω12 =?ω21a127,a52a39
a233a193a123,a79a36a183a162a25a243a139a24a199a92a160a35a37a222,a139a24a199a204a27a37a222,a164a139a24a199a166a199
3
a162a18a134a123a196a24a199a41a37a123e3 = e1 × e2(a37a222a232),a233a149a89a199a41a37,a183a178a199a123a174a199
a48a199a209a233a140a159a70a60a106( a135a73),a210a44a49a178a89a199a60a106a44a143e1,e2,e3 a123a24a199a34a117a28
a92,a89a199a28a92a52a89a248a123a24a28a48a199a213a145a24a199a48a199a52de1 = ke2,a79a143a183a162a123a48a106
a52a39a233a193a123,a204a233a110a34a222a123a195a163a209a520,a196a49ω13 = 0,a98a52a217a198a123a195a163a7a213
a63e3a123a160a152,a132a149e1,e2 a160a10a123a7a248,ω13 a52a127a1490a123,a79a36ω31 a15a127a1490,a196a49
a89a199a48a106a85a241a117,a210a52a183a5a193a1233a199a48a199:
de1
ds = ke2
de2
ds =?ke1 + ωe3 (3.7)
de3
ds =?ωe2
a89a28a48a199a52a104a35a77a34a88a21a48a123a30a127a33a0a123a24a28a48a199,a183a162a47a158a119a199Frenet
a48a199,Fren eta52a27a41a123a24a199a106a166a27,a183a46a89a52a198a123a123a70a44a88a169,a198a88a24a189a52
a62a24a199a201a241a89a199a48a199,a104a30a139a1a199a124a41a241a89a199a211a42,a44a183a89a123,a28a162a44a49a23
a241a117a122a89a199a48a199a63a88a212a104a10,a79a36a183a248a234a77a71a49a105,a164a139a12a24a199a60a106ω,ω
a210a52a48a199a165a123a12a71(torison),a15a52a140a159a123a60a106,a52a44a43a56a45a123a77a34a243a228a196a107
a5a123a117a159,a196a49a56a45a77a34a139a220a199a60a106,a24a199a52a77a71,a12a24a199a52a12a71,a12a71a210a52
a195a85a199a14a143a248a88a52a24a30a178a193a77a34,a199a52a243a56a45a107a77a123a24a199a222,a196a49a56a45a77a34
a52a126a220a199a60a106a117a195a85a123,a199a162a137a219a143a195a85a89a56a45a123a117a159,a89a220a199a60a106a23a108a105
a173a7,a79a143a199a162a7a52a127a1490a123a30a127,a210a44a43a234a77a34a105a59a92a123a117a159,a7a52k = 0
a123a155,a89a77a34a210a196a143a134a34,a89a105a142a52a121a210,a183a5a193a201a241a121a210a213a79a143k = 0,a196
a49de1 = 0,a79a36a92a160a35a37a222e1a52a24a199a158a106,a79a143a89a248a199a123a135a73a98a127a1490,a0a143
a28a178a89a199a158a106e1 = Ca83a156a116e1 = dxdsa165,a242a178a199a232a73a24a5a6a210a122a116xa52s a123a24
a39a60a106a213
x = Cs,(3.8)
a196a49a89a210a52a24a30a134a34,a39a44a117a15a105a142a52a21a121a210a152a42a28a139a24a30a134a34a123a155,a199a123
a77a71k a52a127a1490a123,a79a36k = 0 a83a44a77a34a123a33a59a92a123a117a159,a210a52a134a34,a0a143a7
a213a63a123a52a243a189a12a71a123a30a127,a24a189a7k negationslash= 0,a165a52k = 0 a123,a149a52de1ds = 0,a15a210
a52a134a34a234,a89a30a199a210a150a139a27a6a189a204a27a34,a24a30a134a34a203a199a18a134a123a52a24a199a178a193,
4
a89a199a178a193a176a62a196a139a203a36a134a34a18a134a123a48a53a209a52a139a51a248a123a117a159,a196a49a210a150a139a204
a27a34,a79a36a15a88a21a189a12a71,a12a71a24a189a7a243a89a154a123a77a71k negationslash= 0a123a30a127a98a139a63a66.
a13a104a12a71a127a1490a123a30a127,a104a108a210a52a44a43a89a30a77a34a52a243a178a193a222a24a30a77a34,a5a193
a183a15a201a234a24a199a59a92a123a121a210,a79a143a12a71a89a199a60a106a52a243Frenet a218a42a123a145a174a199a218
a42a176,a196a49a132ω = 0 a44a127a36a30e3a52a199a158a106a123a37a222,a233a149e3 a89a199a158a106a37a222,a132
a149e3a52a24a199a27a34,a63a37a79a143a27a34a203a35a34a52a123a207a18a134a123a199a196a49e3 a203dxds a123a19a232
a52a123a207a127a1490,a79a36e3a7a52a127a149a158a106a123a155,a183a210a44a49a178a48a199
(e3,dxds) = 0 (3.9)
a232a73,a79a143e3a52a24a199a158a106a199a196a49a89a232a73a210a52e3 a203x a123a19a232a127a149a24a199a158a106,a79
a36a199a52a24a199a178a193a77a34,a149a52ω = 0a52a44a43a77a34a52a199a178a193a77a34,a39a44a117,a44a49
a105a142a52a121a210a178a193a77a34a123a12a71a520,a196a49a77a71a90a12a71a220a199a60a106a209a139a59a92a123a1a91
a117a159,a12a105a24a171a105a139a63a135a123a77a34a52ka166ωa209a88a1430,a98a209a52a158a106,a0a143a243a89a199
a60a111a131a5,a44a49a121a210a77a34a52a199a90a34,a210a52a89a248a59a92a123a135a232a73a123a97a126a243a9a212
a154a166a222a139a173a7a123a63a66,a79a143a183a162a127a119a9a212a154a166a123a24a199a204a7a123a154a92a212a52DNA.
DNAa52a220a30a90a34,a52a199a118a90a34,a89a52a9a212a222a58a158a228a253a131a25a54,a243a9a212a222,a89
a248a123a77a34a210a115a241a117a234,a79a36,a77a34a88a243a9a212a154a166a139a173a7a123a97a126,a15a210a52a76a27
a7a127a119a77a34a123a117a159a152a91a107a47DNAa123a154a166a117a159,a196a49a106a166a210a105a173a7a234,a77a34
a123a117a159a107a47a116a154a166a123a117a159,a131a217a52a243a154a166a176a62,a139a30a127a28a178DNAa35a228a234,a199
a123a117a159a210a149a35a234,a196a49a35a228a131a128,a106a166a117a159a210a21a9a149a35a199a199a123a154a166a117a159a15a149
a35,a255a88a199a162a152a91a149a35a199a89a52a243DNAa123a207a196a249a243a9a212a154a166a48a193a52a58a158a228a253
a123a175a11,a76a27a41a234a105a245a123a211a42,a25a243a252a24a253a135a9a212a154a166a123a86a7a32a13a117a123a155,
a210a23a68a139a24a199a228a253a123a218a42,a119a41Whitea218a42,Whitea52a183a123a24a199a166a9a123a166a9,
a198a41a89a199a211a42a52a198a70a44a88a169a123a24a199a136a42,a198a228a237a105a80,a198a89a199a136a42a35a143a9a212
a154a166a123a24a199a228a253a123a218a42,a183a25a243a88a21a89a89a199a136a42.
5
3 Feuchlla88a127a42,Croftona232a73a218a42a90Fary-Milnor
a189a174
a41a89a199a1a91a207a196a123a30a127,a139a74a173a7a117a159a7a255a88,a47a158a173a7a123a117a159a128a128a52a114a14
a1a91a123a117a159,a243a77a34a123a60a89a15a88a52a207a196a24a66a227a123a1a91,a183a162a25a243a178a222a193a123a89
a199a119a41a219a92a123a1a91a199a253a207a196a24a66a227a123a1a91a117a159,a89a30k,ωa209a52a140a159a123a60a106,
a126a199a162a207a196a199a123a117a159,a52a14a143a248a107a77,a88a44a205a173a7a123a52a207a196a114a30a77a34,a34a38
a139a24a30a77a34a52a85a20a123,a23a89a30a77a34a139a31a143a248a123a117a159,a89a176a139a199a58a158a7a155a123a218
a42,a233a149a89a30a85a20a123a77a34,a154a154a139a199a77a71,a79a143a85a20a234,a183a210a207a196a92a77a71,a253
a178a89a199a77a71a215a114a30a77a34a70a199a123a232a73,a149a52a210a139a196a162Feuchlla218a42
integraldisplay
c
kds ≥ 2pi,(3.10)
a210a52a128a89a199a232a73a139a199a5a142,a24a189a522pi,a89a199a134a250a89a229a117a210a52a152a42a28a24a154a77
a71a209a150a139,a89a30a77a34a210a134a34a18a5a86,a52a88a204a85a20a229a117,a28a7a134a34a18a5a86a21a234
a85a20a123a155,a15a7a178a199a107a44a117,a107a44a117a210a7a139a77a71,a0a143a152a42a21a234a107a44a117,a166a198
a117a143a48a233a222a117a123a155,a0a143a77a71a243a114a199a77a34a222a123a232a73a139a24a199a251a189a123a5a142,a89a199
a5a142a522pi,a89a210a52a196a162a123Feuchlla218a42,a183a7a121a210a89a199a218a42,a143a31a143a89a92a77a71
a7a139a24a199a5a142? a121a210a89a199a218a42a123a48a27a210a52a255a88a89a30a77a34a123a92a160a35a37a222,a253
a243a35a34a123a48a53a123a24a199a92a160a37a222,a204a7a10a39a210a52a188a126a176a132a110a243(Gau ss map).
a233a149a89a248a24a199a92a160a35a37a222,a243a251a189a154a82a203a89a199a35a37a222a52a178a113a123a0a74a92a160a37
a222,a196a49a28a139a142a245a92a160a37a222a52e1(s),a199a52a159a127a1491 a123a24a199a37a222,a196a49a28a34a38
a178a199a23a196a56a45a24a199a154,a199a210a52a243a92a160a69a222a123a24a199a154,a79a36a28a82a24a199a92a160a69,
a82e1(s)a23a196a92a160a69a222a123a24a199a154,a0a143a104a28a89a199a154a215a198a117a77a34a18a24a87a123a30a127,
e1(s)a243a92a160a69a222a196a24a30a77a34,a89a52a176a132a110a243a123a63a135,a253a196a162a123a176a132a110a243,a89
a30a77a34e1(s)a243a92a160a69a222a196a196a123a123a77a34a88a52a198a117a123a77a34,a166a234a24a171a234,a79a143a199
a123a48a199a52e1(s),a88a52x(s),e1(s) a119a22:
dσ2
ds2 = (
de1
ds,
de1
ds ) = k
2,σa52e1(s)a123a140a159,(3.11)
a92a160a69a193a152a42a139a24a199a76a203,a0a143a89a199a76a203a139a24a199a244a154,a183a119a199y,a15a210a52a128,
6
a76a203a52a6a204a203,y a52a89a199a76a203a123a244a154.a178a89a199a244a154a104a196a186a222a123a24a154,a0a143a77a34
a210a139a199a176a221,a89a199a176a221a250a82a196a52y(x),a52a199a19a232,a89a199a176a221a44a49a23a196a52a77a34
a222a123a24a199a60a106,a0a143a77a34a243a56a45a176a62a210a139a234a24a199a176a221,a183a162a89a77a34a52a85a20a123,
a104a108a139a199a33a176a154,a139a199a33a132a154,a243a33a176a154,a79a143a199a52a33a176a123,a23a108a199a123a35a34a52
a178a123,a243a33a132a154,a35a34a15a52a178a123,a196a49a210a139a220a199a248a142a154,a24a199a52a83a44a176a221a33
a176a123,a24a199a52a33a132a123,a13a166ya89a199a154a18a134a123a0a199a76a203,a24a189a203a92a160a69a222a123a77a34
a35a98,a79a143(y,dx) a89a199a19a232a520,a196a49ya203a69a193a123e1(s)a243a24a154a35a98,a79a36a243a92
a160a69a222a123a77a34a203a89a199a92a160a203a35a98,a98a52ya52a127a63a48a53,a196a49a183a162a69a222a123a176a132
a110a243a123a77a34a52a203a69a222a123a31a143a203a209a35a98,a89a52a79a143ya52a127a63a24a199a154,a149a52ya123a0
a199a6a204a203a98a183a162a89a30a77a34,a183a46a176a132a77a34a183a119a199γ,a199a203ya123a6a204a203a35a98,a79
a36a203a127a63a92a160a203a35a98,a89a52a105a7a155a123a24a199a117a159,a196a49γa89a30a77a34a24a189a7a35a104
a159,a199a243a69a222a88a52a24a30a127a91a123a77a34,a199a243a69a222a52a24a30a203a196a139a123a203a177(a76a203)a209
a35a98a123a77a34,a0a143a183a164a52a137a7a121a210a199a1392pi a89a199a5a142,a5a193a117a128a210a89a199a175a11.
a183a89a30a77a34γa52e1(s),a119a51a160a203,e1(s) a210a52a56a45a77a34a123a24a199a176a132a110a243,a199a52
a35a34a123a48a53a243a92a160a69a222a196a196a123a77a34,a13a89a30a77a34a119a41γ,a199a203a69a222a123a127a91a92
a160a203a209a35a98,a0a143a24a196a143,a89a30a77a34a35a104a159a234,a152a42a212a225,a199a150a139a89a199a117a159.
a128a199a139a24a1992pi a123a5a142,a52Croftona218a42a213
integraldisplay integraldisplay
= 4L,(3.12)
Croftona123a169a57a15a105a139a63a135,a88a52a24a199a116a42a123a106a166a169a57,a93a41a123a186a41a92
a86(Encyclopaedi a)a62a198a85a24a159a169a57,a52a7a149a1a91a150a71a123,a198a52a243a85(a1a91)a150
a71a123a169a57a30a127a178a198a123a136a42a85a243a176a62a234,a89a199a136a42a105a7a155,a166a233a155a128,a210a243a178
a193a222a89,a77a34a123a159a221a44a49a44a143a203a199a35a98a123a134a34a123a221a222,a159a221,a210a52a134a34a123a221
a222,a89a199a63a135a243a230a212a166a176,a163a83a243a26a166a209a139a97a126,a139a30a127,a28a252a14a222a123a192a220,
a7a175a199a139a245a76,a15a88a21a178a124a35a13,a52a88a80a222a123,a149a52a210a52a126a23a35a98a89a199a192a220
a123a34a42a143a221a222,a117a222a89a199a252a14a222a123a236a24a92a73a123a76a66,a89a24a92a73a106a166a24a196a119a41
a232a73a1a91,a88a52a135a73a1a91,a232a73a1a91a210a52a207a196a89a74a232a73a123a7a248a166a117a159,a232
a73a1a91a243a26a166a222a139a105a76a123a97a126,a139a105a245a229a236a102a126a102a126a89a174a88,a79a143a28a7a23
a124a123a62a14,a15a88a21a252a89a199a62a14a117a222,a210a252a34a243a199,a149a243a199a123a72a42a117a23a62a14a123
7
a76a66a249a217a199a123a117a159,a89a210a52a232a73a1a91,a0a143a183a25a243a128,(3.12)a52Croftona218a42,
a210a52a128a69a193a222a30a15a139a1a91,a52a69a193a1a91,a199a123a154a183a209a127a119,a199a123a134a34a210a52a76
a203,a196a49a25a243a183a243a69a193a222a139a30a77a34,a203a196a139a76a203a209a35a98,a0a143,a89a171a134a34a123a199
a106a139a24a199a222a221,Croftona218a42a89,a199a52a127a1494a245a199a123a222a221,Feuchlla218a42a52a69a193
a222Croftona218a42a123a24a199a136a42,a79a143Croftona218a42a128a199a123a222a221a524a245a149a199a123a143a30
a52a159,a196a49a210a188a126a89a30a77a34a203a154a24a199a76a203a209a35a98a123a117a159a210a122a116Feuchlla218a42.
a79a36a175a11a210a52a128a31a143a52a69a193a222a76a203a123a222a221,a69a193a222a123a76a203a52a69a193a222a58a78a1
a91a123a134a34,a13a233a149a134a34,a183a15a139a199a222a221(measure).a243a69a193a222a123a222a221a105a59a92,a89
a52a79a143a69a193a222a123a134a34a203a244a154a139a199a59a92a123a233a83a7a248,a79a143a28a139a24a199a76a203,a199a139
a24a199a244a154,a79a36a89a30a134a34a203a89a199a244a154a139a199a233a83a7a248,a149a52a89a199a76a203a123a222a221a210
a82a143a89a199a244a154a123a222a221,a196a49a183a25a243a76a150a89a24a89a14a143a248a121a210a69a193a222a123Crofton
a218a42,a89a199a121a210a217a34a105a59a92,a88a44a7a66a101a24a154,a210a52a46a27a6a166a166a43a41a210a113a234.
a25a243a7a48a128,γa89a30a77a34a203a24a199a76a203a35a98,a0a143a89a199a76a203a44a49a166a43a41,a152a42a166
a43a41,a132a149e1(s)a89a30a77a34a203a89a199a76a203a35a98,a13a76a203a139a199a244a154y,a196a49a7a166ya123
a43a41,a0a143a166a43a41a166a31a143a17a218a89a199a76a203a166a134a34a35a98,a13a76a203a522a145a123a56a45,a196
a49a7a139a220a199a43a41,a183a210a82a89a30a77a34a123a140a159sa42a143a24a199a43a41,a0a143,a89a199a76a203a203
a134a34a35a98a123a155,a210a139a67a222a123a89a199a60a89a213a164a98a24a199a43a41a52a31a143a17a218a76a203a139a24
a199a244a154,a89a199a244a154a52y,a23a108ya52a166e1a18a134a123,a196a49y a24a189a243e2,e3 a196a196a123a203a177
a222,a89a52a79a143e2,e3a52a203e1 a18a134a123,a79a36ya210a52a243e2,e3a123a203a177a222,a149a52ya210a44a49
a44a143
y = cosθe2(s) + sinθe3(s),(3.13)
a34a19a222,ya52e2,e3 a123a24a199a34a117a28a92,a145a79a143a199a52a199a92a160a37a222,a196a49a44a49a85a196
a89a248a123a111a231,a79a36ya89a154a139a220a199a43a41,a183a44a49a82a143s,θ,a0a143a25a243a123a175a11a52a70y
a243a89a199a193a232a195a163(element area)a123a221a222,a52a7a178y a89a154a123a221a222a23a196a52a76a203a123
a221a222,a47a158a89a88a10a70,y a52a24a199a154,a28a210a70dy,ya52a243a92a160a69a222a123a24a154,a149a52a5
a193a117a70dy,a178a89a199dya85a143a203y a18a134a123a220a199a48a53a123a60a106,a0a143a89a220a199a92a160a37a222
a248a106a123a105a232a210a52y a123a221a222,a196a49a183a25a243a89a143a41,e1,e2,e3a52a41a37,a199a162a174a199a37
8
a222a52a144a35a18a134a123a92a160a37a222,a199a162a15a209a52sa123a60a106,a196a49a183a178a199a123a218a42a85a241a117:
de1
ds = a2e2 + a3e3
de2
ds =?a2e1 + a1e3 (3.14)
de3
ds =?a3e1?a1e2
a195a88a152a91,a89a218a42a176a62a123a248a106a196a24a199a39a233a193a123a48a106,a89a210a41a5a86,a105a59a92a143,
a152a42a174a24a174dy,a210a122a116dya24a199a218a42:
dy = (?sinθe2 + cosθe3)(dθ + a3ds)?e1(a1 cosθ + a3 sinθ)ds,(3.15)
a149a52a183a162a139
a193a232a195a163= (a2 cosθ + a3 sinθ)dθ ∧ds,(3.16)
a165a215
a2 = cosτ,a3 = sinτ,(3.17)
a11
a193a232a195a163= cos(τ?θ)dθ ∧ds,(3.18)
a253a33a128cos(τ? θ)dθ ∧ ds a52a69a193a222a123a193a232a195a163,a143a234a7a70a89a199a69a193a222a123a193
a232,a70a89a199a192a220a123a173a232a73,a15a210a52a70a89a199a42a6a123a173a232a73,a0a143a183a106a89a199a154a123
a30a127,a52a31a68a113a92a253a233a138,a117a70a89a199a253a233a138a123a89a199a173a232a73,a183a166a98a34a247a139a220
a199a35a106s a203θ,a0a143a210a233a199a162a117a70a232a73,a232a73a123a30a127a11a251a189s,a82cosa123a253a233a138
a117a70a232a73,a34a38a110a221a221a24a87a123a155,|cos|a24a225a35a234a245a232a17a218cosa123a253a233a138a123a232
a73a524,a89a52a79a143a104cosa440a116pia30a52a441a116-1,a24a225a524a391,a196a49a524,a79a36a89
a248a28a210a70a116a203a77a34a35a98a123a76a203a123a221a222,a253a139a245a232a199a76a203,a89a199a221a222a52a70a89a199
a173a232a73,a178a199a174a241a117,a174a241a117a131a128,a28a144a7a70a24a39a233θa123a232a73a210a44a49a234,a16a5
a123a52a77a34a123a140a159,a196a49Cr oftona175a11a210a21a44a89a199a15a174a122a116,a28a162a168a5a117,a195
a86a46a24a46,a210a44a49a122a116,a79a36Croftona218a42a128γa89a30a77a34a150a232a139a245a159,a199a178a89
a199a77a34a203a127a91a76a203a35a98a123a117a159a44a143a24a199a221a222a123a117a159,a253a139a245a159,a132a89a199a183
a210a122a116Feuchlla189a174,a89a52a105a164a224a123a24a199a189a174,a183a162a34a38a89a199a22a111a52a85a20a123,
9
a2432a145a123a60a111,a210a52a85a20a123a77a34,a233a149a85a20a123a77a34,a199a123a15a77a71a139a24a199a24a189a123
a5a33,a31a143a30a127a89a199a5a33a21a72a116? a23a108a104a89a30a77a34a52a178a193a77a34a63a37a52a24a30
a113a247(complete)a77a34a30a52a44a49a72a116a123,a202a226a222a193a255a88a15a44a49a121a210a89a199a136a88.
a205a139a63a135a123a24a199a136a42a52a196a162a123Fray-Milnor a189a174a213a152a42a77a34Ca139a136,a11
integraldisplay
C
kds ≥ 4pi,(3.19)
a183a46Milnora52a163a74a35a117a155a41a33a128a132a123a24a199a95a192a166a27a86,a198a41a89a199a189a174a123a30
a127,a210a203a28a162a24a248,a222a49a30,a144a19a225a116a89a199a175a11,a198a201a241a24a199a30a71a31a143a30a127
a24a30a85a20a123a77a34a21a75a196a24a199a136,a34a38a139a136a123a155,a23a108a183a162a77a127a137a7a77a71a205a245
a24a74,a79a143a75a136a210a122a221a221,a88a44Fray -Milnora189a174a210a52a89a34a38a89a199a85a20a123a77
a34a139a24a199a136a123a155,a199a123a92a77a71a123a232a73a150a232a1434,a196a49a92a77a71a24a189a210a76a1494,a89
a199a121a210a105a59a92,a79a143a152a42a199a66a1494a123a155,a24a189a139a199a48a53,a243a89a199a48a53a222a123a176a221
a144a139a24a199a244a76a203a24a199a244a66a154,a196a49a183a89a30a77a34a24a189a52a203a122a89a199a244a76a154a166a244
a66a154a123a220a199a140,a0a143a165a45a210a150a139a244a76a154a90a244a66a154,a196a49a165a45a0a74a178a113a123a178a193
a203a77a34a35a98a123a155,a209a35a98a149a220a154,a0a143a126a24a30a34a227a178a89a220a154a203a229a117,a149a52a199
a89a77a34a210a140a196a24a199a75a173,a23a108a77a34a210a44a49a192a196a24a154,a196a49a210a88a52a136,a196a49a152
a42a89a92a77a71a66a1494 a123a155,a199a89a30a77a34a123a136a210a44a49a137a13,a79a36a152a42a77a34a139a199a136
a123a155,a199a123a92a77a71a24a189≥ 4,a149a52a193a47a122a241a117Fray-Milnor a189a174,a5a193a183a89a89
a166a189a41a98a123a211a42a123a7a248,a189a41a98a123a211a42a52a255a88a89a199a136,a159a113a88a144a24a30a77a34,
a44a49a139a80a1a30a77a34,a253a196a162linka123a207a196,a255a88a31a143a30a127a89a199link a21a119a44a117a77
a71a52a12a71,a89a199a175a11a212a174a166a27a105a161a108a85,a33a163,a41a234a105a245a89a48a193a123a211a42,a88
a44,a77a18Fary-Milnor a189a174a139a199a44a21a117,a210a52a255a88a35a34a123a77a71a203a12a71a233a149a189
a41a98a123a88a35a42a139a31a143a107a47,a189a41a98a52a207a196a77a34a123a233a83a51a174a123a117a159a203a135a73a42
a123a7a248,a196a49a93a116a126a243a89a199a48a193,a28a88a68a122a21a178a89a199a136a137a13,a98a52a139a44a21a104
a77a34a123a77a71a166a12a71a139a236a171a117a159a123a30a127,a198a196a189a123a89a199a88a35a42a204a127a1490,a196a49
a139a74a175a11a207a196a123a30a127a44a49a46a24a46,a152a42a139a124a38a234Harrisa115a71a123a49a123a155,a198a210
a243a89a189a41a98a123a211a42,a189a41a98a52a163a1a35a117a165a41a151a9a123a24a199a58a158a128a132a123a106a166a27.
a198a150a139a214,a88a209a166a152a144a139a31a143a7a248,a24a199a124a41a198a123a11a173,a41a58a158a1a79,a58a158a13
a99a123a211a42.
10
a145a143a89 a77a193a88(a24)
2001a3511a2192a134
1 a77a193a123a41a37
a152a21a183a162a89a24a154a77a193a88,a135a232a73a243a77a193a222a97a126a123a207a196a243a114a199a106a166a176a62a52a105
a7a155a123,a89a52a79a143a243a77a193a88a165,a77a193a123a89a74a117a159a128a128a106a223a116a217a198a205a18a123a60
a111,a13a89a74a205a18a123a60a111a35a154a116a77a193a123a30a127a15a139a105a245a117a159,a243a77a193a123a60a111a46a178
a21a9a234,a0a143,a77a193a139a199a128a154,a210a52a183a162a34a189a199a52a2433a145a56a45a176a62,a196a49a28a23
a122a68,a28a44a49a152a67,a44a49a243a23a122a68a199a222a62a123a77a34a176a123a117a159a249a217a198a31a143a123,a24
a116a176a145a49a128,a210a23a88a68a234,a183a123a89a27a203a86a88a24a248,a196a49a183a46a76a27a178a89a1a16a97
a238a133a92a24a5,a89a199a97a238a76a150a97a148a243a202a47a135a73a1a91a86a222a73a88a116a123,a199a139a199a128
a154,a210a52a79a122a105a13a37a48a27a117a122a59a92,a0a143a31a143a52a77a193a17? a77a193a210a52a67a222a24
a199a59a77a123a192a220,a183a178a199a123a154a123a43a41a44a143a220a199a35a106a123a60a106,a89a220a199a35a106a183a119
a41u,v,u,va24a196a119a41a107a106(parameter),a34a38u,va35a154a123a30a127,a89a74a154a123a27a236
a210a196a234a24a1992a145a123a77a193x(u,v) = (x1(u,v),x2(u,v),x3(u,v)),a149a52a79a143a139u,v,
a196a49a28a44a49a38a122a24a199a107a106a123a138a52a158a106,a108a128a38a122a12a24a199a107a106a35a154,a247v a52
a158a106,a13u a35a154,a196a49a210a139a24a30a77a34,a199a123a107a106a52u,a51a248,a28a44a49a38a122ua88
a35,a13va35a154,a79a36a243a77a193a176a62,a139a220a28a77a34,a199a123a107a106a24a28a52v a127a149a158a106,a24
a28ua52a127a149a158a106,a233a149a89a220a28a77a34,a154a24a30a77a34a243a154a24a154a209a139a24a30a35a34,a196a49
a243a24a199a154x,a183a162a210a139a220a30a134a34,a183a34a189a89a220a30a134a34a88a173a92,a166a233a155a128,a137a219
a143a89,xu,xv a88a52a51a24a199a48a53,a88a178a134(a34a117)a35a7,a217a165xu,xva210a52xa123a37a222a73
a47a233u,va70a160a135a73,a221a86a183a128,a199a123a37a222a232xu ×xv negationslash= 0,a34a38a89a220a199a48a53a88a173
a92,a196a49a199a162a210a60a196a24a199a178a193,a89a199a178a193a183a119a41a77a193a243a89a24a154a123a35a193,a89a199
a35a193a243xa89a24a154a139a199a18a134a123a48a53,a89a199a48a53a123a134a34a24a196a119a41a27a34,a215a248a27a34
a123a48a53a139a24a199a92a160a37a222,a79a36a15a119a41a27a37a222,a89a199a27a37a222a139a220a199a160a10,a199a44
a49a53a222a18,a15a44a49a53a5a18,a139a220a199a48a53a166a80a35a39a123a160a10,a183a160a10a199a38a122xu,xv
a203a27a37a222a52a24a199a141a67a123a43a41a41a37,a52a24a199a141a67a248,a166a233a155a128,a183a119a89a199a27a37
1
a222e3,a63a34a247e3a52a199a92a160a37a222,a149a52e3a210a119a22a30a71
(e3,e3) = 1 (a92a160a37a222),e3 = xu ×xv|x
u ×xv|2
,(4.1)
a0a143a243a89a248a123a160a10a131a5,(xu,xv,e3)a210a52a141a67a248,a89a30,a113a239a42det(xu,xv,e3)a52
a116a123,a253det(xu,xv,e3) > 0,a196a49a52a141a67a248,a25a243,a183a123a89a199a48a27a203a24a196a123a48
a27a52a88a51a123,a24a196a123a86a222a128a128a126u,v a107a106a21a48a114a199a123a77a193a123a135a73a1a91,a79a36
a210a7a118a159a234,a198a162a89a176a139a24a199a98a154,a79a143xu a203xv a88a24a189a18a134,a0a143a183a162a123a108
a85a52a243a149Euclid a1a91a139a24a199a221a222,a196a49a198a162a126a123a52a58a18a134a123a43a41a248,a13a1a91
a52a24a248a123,a98a52a73a219a48a193a123a218a42a210a7a118a133a236a234,a13a183a82e1,e2a52a92a160a37a222,a199
a162a52a144a35a18a134a123,a196a49a25a243e1,e2,e3a174a199a37a222a209a52a92a160a37a222,a13a37a144a35a18a134,
a63a37a79a143a7a199a52a24a199a141a67a248,a196a49a199a123a113a239a42a97a148a52a116a123,a98a52a79a143a89a174a199
a37a222a209a52a144a35a18a134a123a92a160a37a222,a196a49a113a239a42a127a149±1,a199a123a178a48a127a1491,a196a49
a183a25a243a52a119a89a199a113a239a42a127a1491 a123,a79a36a89a52a24a199a116a98a123a43a41a248,a199a123a113a239a42
a127a1491,a149a52
e1,e2,e3 (a141a116a98a41a37),(ei,ej) = δij,(e1,e2,e3) = 1,1 ≤ i,j,k ≤ 3,(4.2)
2 a77a193a123a135a73a42a249a217a1a91
a0a143a135a73a1a91a14a143a248a17? a89a30a210a88a144a52a139a24a199a43a41a248,a13a52a139a24a24(family)a43
a41a248,a164a139a1a199a35a106(a35a123a107a106),a0a143a33a7a155a123a24a199a25a54a210a52a24a199a43a41a248a203
a199a248a163a123a43a41a248a52a14a143a24a199a7a248,a7a234a137a89a199a7a248a52a135a73a1a91a33a204a7a123a175
a11,a196a49a183a25a243a139a24a24a43a41a248,a7a48a128a52a139a220a199a35a106u,v,a4a150a44a49a139a245a199a35
a106,a0a143a7a73a89a199a248a163a43a41a248a203a199a123a7a248,a183a210a178a199a135a73a234,a25a243a183a89a199xa52
a37a222,eia15a209a52a37a222,a196a49a183a210a70a70a23dx,a23da203dei,a89a52a24a199a37a222a123a135a73.a98
a52a79a143e1,e2,e3 a52a24a199a41a37,a52a34a117a195a7a123,a13a183a162a52a2433a145a56a45a165a255a88a123,
a196a49a127a91a24a199a37a222a23a108a52e1,e2,e3 a123a34a117a28a92,a196a49a183a44a49a178deia85a196
dei = ωijej,(4.3)
2
a89a176a183a126a123a52Einsteina123a110a82,a152a42a139a24a199a141a106a173a133a123a155a210a35a28,a79a143a183a162
a123a56a45a523 a145a123,i,j,ka209a52a441a1163,a0a143ωijej a210a52a233j a35a28,a196a49a89a523 a49,
a253j = 1,2,3,a1393a49,a89a52Einstein a243a135a73a1a91a90a159a123a110a82,Einsteina164a41a234
a24a71a47a60,a7a48a128,a44a3a28a7a139a199a106a248,a199a7a139a199a141a106,a253xi,ia89a199a141a106a209a85
a196a5a41,Einsteina128a88a85a5a41,a198a210a85a234a222a41,a79a36a128a117a123a135a73a1a91a123a86a176
a62,a222a41a58a158a245,a43a41a123a89a199a141a41a209a85a196a222a41,a217a34,a222a5a150a139a7a248,a196a49,
a183a126a5a41a85a196a218a42(4.3),a218a42(4.3)a52a228a253a123a218a42,a199a44a43a220a199a248a163a43a41a123
a7a248,a24a199dei a203a198a117a123a43a41ei a123a7a248,ωija52a31a143a17? a199a52a24a39a135a73a42,a79
a143dei a52a24a39a135a73a42,a199a123a138a52a37a222a123a24a39a135a73a42,a196a49a28a152a42a85a196a199a123a73
a222a123a155,a210a1393a199a73a222,a13a37a154a24a199a209a52deia123a73a222,a0a143ωija52a24a39a135a73a42,a79
a143i,ja209a52a441 a1163,a196a49a24a225a1399a199,a98a52a89a74ωij a88a52a127a63a123,a89a52a1a91a222,
a197a166a222a33a228a253a123a24a199a218a42,a79a143a183a162a24a35a209a52a116a98a123a248a58,a196a49ωija233a149a5
a41i,ja52a39a233a193a123,a253
ωij + ωji = 0,(4.4)
a143a31a143a17? a79a143ei,ej a119a22(ei,ej) = δij,a28a178a199a135a73,a145a79a143 deltaij a52a158a106,
a196a49a135a73a199a131a128a520a199a149a52a210a21a122a116a5a193a218a42:
(dei,ej) + (ei,dej) = 0,(4.5)
a132a149dei = ωikek,a63a37ei,eka52a144a35a116a98a123,a196a49a132a222a42a210a122a116ωij+ωji = 0,a253
a218a42(4.4),a233a149a196a139a123i,ja441a1163,ωija52a39a233a193a123,a79a36(ωij)a23a196a24a1993×3a123
a48a106,a0a143a89a199a48a106a52a39a233a193a123,a199a88a52a1399a199a195a163,a34a19a222,a144a1393a199,a63a37
a79a143a39a233a193,a196a49a204(a233a110)a34a222a123a195a163a209a520,a217a198a123a233a149a204a34a52a39a233a193a123.
a15a210a52a139
(ωij) =
0 ω12 ω13
ω12 0 ω23
ω13?ω23 0
,(4.6)
a79a36a144a16a53a199a195a163,a253a144a139ω12,ω13,ω23,a893a199a209a52a24a39a135a73a42,a13a893a199
a24a39a135a73a42a233a77a193a1a91a117a159a52a58a158a7a155a123,a253a199a162a209a44a49a132a135a73a42a117a44a43.
a202a47a207a196a60a106a88,a180a73a219a123a30a127,a209a52a89a60a106,a255a88a60a106a221a86a52a178a60a106a135a73
3
a234,a178a199a123a135a219a42a143a248a106,a76a27a4a58a88a243a14a149a24a39a135a73a42,a217a34,a24a39a135a73
a42a52a178a89a199a175a11a65a122a59a92a234,a183a13a62a123a30a127a17a178a203a28a162a144a235a44a27a41a123a76a106
a166a27Darbouxa123a86,a199a139a143a76a253a119a41a21a77a193a88a22,a89a171a86a105a138a122a23,a88a44a44
a234a52a27a169a123,a198a88a126a135a73a42,a198a126a123a52a160a135a73,a196a49a139a142a245a218a42a85a229a117a159a24
a74,a126a135a73a42a123a155,a24a39a135a73a42a85a229a117a210a59a92a245a234,a196a49a183a89a176a126a234a24a39a135
a73a42,a126a116a98a41a37,a38a77a193a88a58a158a59a92,a89a74a28a162a243a202a47a135a73a1a91a86a165a105a232
a21a73a116,a98a52a89a171a48a27a105a139a72,a79a143a24a35a192a220a209a59a92a234,a183a34a189a77a193a52a189a53
a123,a253a243a221a123a30a127a139a24a199a39a30a168a48a53,a79a143a189a234a53a131a128,a243a24a199a154,a199a139a220a199
a35a37a222a123a155,a199a123a27a37a222a210a113a92a104a189a234,a79a36,a77a193a189a53a131a128,a154a24a154a24a189a139
a24a199a251a189a123a92a160a27a37a222,a88a52a199a123a139a123a37a222,a0a143,a77a193a222a139a24a199a105a7a155a123
a1a91a136a232,a210a52a24a199a154a28a24a199a92a160a35a37a222,a253xa203e1,a89a52a25a83a196a162a14a145a45a123
a33a59a92a123a60a89,a15a52a33a7a155a123a60a89,a0a143a233a149x + e1,a199a245a234a24a199a145a106,a79a143
a251a189a234xa131a128,e1a89a199a92a160a35a37a222a164a44a49a221a87,a196a49xa123a27a236a522a145a123,a0a143a154
a24a199xa123a35a37a222a164a122a281a145,a89a52a79a143a199a44a49a52a24a199a203,a221a24a199a203a177,a199a52a92
a160a123,a196a49a89a199a56a45a523a145a123,a13a89a1993a145a56a45a183a119a41E:
E = {xe1|x ∈ M}(a203a45); dimE = 3,(4.7)
a139a24a1993a145a123a56a45a221a86a128a5a241a24a1993a145a56a45,a89a199a10a39a7a155a244a234,a25a243a142a245a106
a166,a212a174a209a137a7a89a199a10a39,a24a96a28a126a198a117a123a22a111a117a195a85a1a91a88a234,a128a128a137a7
a222a193a139a24a199a203a87,a89a199a183a162a119a41a14a145a45,a89a30a203a177a52a24a199a14a145,a79a36a89a199a14
a145a45a119a41a203a45,a79a143a14a145a52a203,a196a49Ea52a132a22a111M(a77a193a183a119a199a143M,a52a24a199
a22a111)a5a241a117a123a24a199a203a45,a0a143a243a24a154a7a139e1,a210a139e2a234,a89a52a79a143e2 a52a203a199
a18a134a123a24a199a92a160a35a37a222,a51a30a44a49a132a198a117a123a189a53a104a189a5a117,a79a36e1,e2a210a189
a5a117a234,e3a52a27a37a222a104a108a15a189a234,a196a49xe1a89a199a92a160a35a37a222a203a41a37a52a24a195a47.
a139a234a92a160a35a37a222a15a44a49a232a5a24a199a41a37,a104a108a139a234a41a37,a28a210a44a49a82a145a24a199a35
a37a222a143e1,a196a49a893a145a56a45a210a52a183a162a123a77a193a196a139a89a74a41a37e1,e2,e3 a196a196a123a56
a45,a149a52a183a162a210a139a222a193a123a218a42(4.7),a183a128Ea523a145a123,Ea139a24a199a110a243,a110a116a198
a117a123a77a193M,a79a143a28a139a89a199a35a37a222,a199a139a24a199a198a154x,a132xe1a178a199a110a143x,a89a52
a24a199a44E a116Ma123a110a243,a7a207a196a77a193a123a135a73a1a91,a92a44a77a193a88a234,a24a189a7a126E.
4
Ea203a198a117a123a77a193a139a183a35a123a7a248,a166a98a183a89a234a89a171a1a91a123a7a248,a126Ea123a80a255a243
a149,E a123a56a45a523a145a123a56a45,a199a222a62a139a24a39a135a73a42,a89a74a135a73a42a243Ea222a62a209a52
a104a189a123,a0a143a248a234a183a89a123e1,e2,e3a89a1a199a92a160a37a222,a104a108a77a193a123a154xa15a52a24a199
a37a222,a199a123a160a152a37a222a15a52a24a199a37a222,x a52u,va123a60a106,dx a104a108a52u,v a123a24a199
a24a39a135a73a42,a199a15a44a49a44a143e1,e2,e3 a123a24a199a34a117a28a92,a98a52a34a19a222,a199a24a189
a52e1,e2 a123a28a92,a89a52a79a143e3 a52a27a37a222,a196a49a199a24a189a52a243a35a193a222a62,a13e1,e2a209
a52a35a37a222,a196a49dxa24a189a52e1,e2a123a34a117a28a92,a199a123a248a106a183a119a41ω1,ω2,a253a122a116a183
a162a145a24a199a218a42:
dx = ω1e1 + ω2e2,(4.8)
a196a49a183a25a243a1395a199a24a39a135a73a42,ω1,ω2,ω13,ω23,ω12,a89a28a135a73a42a58a158a7a155,a199
a162a209a139a59a92a123a1a91a63a66,a89a199a210a128a210a135a73a42a255a88a1a91a117a159a38a122a175a11a59a92a13
a37a142a52,a25a243a183a162a210a139a218a42( 4.8),a183a162a82a218a42(4.3),(4.4)a42a143a183a162a123a145a19a199
a218a42,a183a243a3a193a89a44,a135a73a42a139a199a33a76a123a128a154,a210a52a135a73a220a39a128a520,a253a233
a149dx,d(dx) = 0,a233a149a127a91a60a106a123a105a135a73a220a39a24a189a127a1490,a89a210a35a104a149a243a56
a45a127a63a82a24a199a75a173,a242a82a199a123a30a142,a13a30a142a88a242a139a30a142,a82a30a142a82a220a39a24a189
a52a127a1490,a79a143a89a199a117a159,a89a171a106a166a136a232a210a139a196a162a123a51a174(homology)a117a159,a25
a243a28a7a180a31a143a192a220,a209a52a51a174a117a159,a58a158a7a155,a88a44a183a162a25a243a178a218a42(4.8)a38
a30dx a242da24a5a6a210a127a1490,a13a178a141a30a123a48a13a210a122d( omega1e1) + d(ω2e2),a253
d(ω1e1) + d(ω2e2) = 0,(4.9)
a213a63a104a105a135a73a3a193a139a24a199a24a39a123a155,a135a73a145a19a199a79a6a7a149a82,a15a13a211a131,a28
a210a178a199a162a135a73a234,a135a73a131a128,a210a21a25a196a122a116a123a42a6a52e1,e2,e3a123a34a117a28a92,a0
a143a199a123a248a106a52a19a39a135a73a42,a13a233a149a89a199a248a106a52a19a39a135a73a42a123a37a222a7a127a1490a123
a155,a196a139a123a248a106a209a7a127a1490,a149a52a28a152a42a13e1,e2a123a248a106a1430 a123a155,a210a122a116a183a5
a193a123a145a174a199a218a42:
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(4.10)
a183a88a45a251a178a199a41a234,a89a199a121a210a105a59a92,a44a193a253a44a122,a89a52a7a155a244a234a123a24a199a218
a42,a89a74a218a42a28a162a15a142a250a122a99,a79a143a243a202a47a86a222a23a88a68,a89a52a132a149a202a47a88
5
a245a161a126a135a73a42,a142a245a124a15a88a204a126a135a73a42,a217a34a89a105a59a92,a196a49a183a210a122a116a218
a42(4.10),a108a128a13e3a123a248a106a1430,a210a122a116a145a143a199a218a42:
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(4.11)
a89a52a58a158a58a158a7a155a123a218a42,a132a89a74a210a122a116a196a162a123Levi-Civitaa178a113a117,a25a243,
a89a199a15a119a201a100(connection),a233a149a201a100,a202a47a73a24a253a86,a44a49a89a222a105a200a105a200,
a217a34a105a59a92,a183a128a243a89a199a60a111a131a5,Levi-C ivitaa201a100a210a52ω12a89a199a24a39a135a73
a42,a213a63ω12 a52Ea176a62a123a24a199a24a39a135a73a42,a199a210a189a1a91a123a117a159,a38a122a183a44a49a178
a89a199a37a222a215a248a24a30a77a34a178a113a123a35a196,a89a52a31a143a63a135a17? a210a52ω12a89a199a24a39a135
a73a42a132a48a199(4.10)a113a92a104a189,a79a143a152a42a139a24a199ωprime12,a38a122a55a92a51a248a123a48a199a42,
a253
dω1 =?ω2 ∧ωprime12; dω2 = ω1 ∧ωprime12,(4.12)
a183a137a7a121a210ω12 = ωprime12,a253a144a139a24a199a44a21a117,a144a139a24a199ω12 a55a92a48a199(4.10),a15
a210a52a128,a34a38a139a12a105a24a199ωprime12a55a92a48a199(4.10),a0a143a183a162a178a220a199a48a199a35a62,a210
a122a116
ω1 ∧(ωprime12?ω12) = 0; ω2 ∧(ωprime12?ω12) = 0,(4.13)
a13a220a199a24a39a135a73a42a152a42a35a198a127a1490a123a155,a13a37a152a42a217a165a24a199a88a520,a0a143a217
a199a0a199a23a108a52a199a123a245a106,a89a52a105a83a106a33a59a92a123a192a220,a28a162a174a24a174a210a241a117
a234,ω1,ω2a52a88a1430 a123,a13a37a88a98a88a127a1490,a63a37a34a117a195a7,a79a36,ωprime12? ω12a17
a127a149ω1a123a245a106,a145a127a149ω2a123a245a106,a98a52ω1,ω2 a34a117a195a7,a196a49a199a7a127a1490.
a79a36a152a42a242a139a24a199ωprime12a55a92a51a248a123a48a199,a199a24a189a127a149?mega12,a89a52a113a92
a104a189a123,a0a143a183a126a89a199a90a159a196a162a123Levi-Civitaa178a113a117,a183a162a25a243a139de1 =
ω12e2 +?mega13e3,a183a162a178e3a89a24a49a82a62,a82a62a52a31a143a63a135a17? a34a38a139a24a199
a37a222,a28a178a199a123e3a82a62a131a128,a210a52a215a248e3a123a48a53a82a89a199a116a98a61a107,a196a49a28a178a199
a82a62a234a123a63a135a210a52a82a199a123a116a98a61a107,a28a178a199a82a62a123a155,a183a25a243a88a242a52a202a47a123
a135a73a234,a52a99a123a135a73a234,a183a126Da44a43a89a199a99a123a135a73,a149a52a210a122a116a89a28a218a42:
De1 = ω12e2,De2 =?ω12e1,(4.14)
6
a89a52a76a183a166a98a121a210a123a52a132a1a91a113a92a104a189a123,a89a52a79a143ω12a52a113a92a104a189a123,a34
a38De1 = 0a123a155,a183a210a128a89a199a37a222a52Levi-Civitaa63a66a5a178a113,a253a183a128a243a89a171a60
a111a5a199a210a52a178a113a123,a199a52a178a113a123,a89a52a139a99a123a63a66,a243Euclid a1a91a123a30a127,a89
a210a52a202a47a178a113,a25a243a52a202a47a178a113a123a77a18,a15a210a52De1 = 0a30a52Levi-Civitaa178
a113,a196a49a243a77a193a222,a28a126a24a39a135a73a42,a166a11a89a199a178a113a7a118a10a195,a13De1 = 0a123
a155,a233a149a37a222a52a199a135a73a48a199,a89a199a135a73a48a199a88a24a189a139a137,a98a52a34a38a28a139a24a30
a77a34,a28a210a44a49a215a248a89a30a77a34a70a137,a233a149a77a34a117a128,a196a139a123a89a74a60a106a209a52ta123
a60a106,a215a248a89a30a77a34a70a137,a44a49a70a116a137,a150a149a89a199a30a71,a183a128a89a199a37a222a210
a52Levi-Civita a178a113,a196a49a44a89a199a189a66a44a49a234a137a89a199a178a113a117a203a77a34a123a160a10a139
a7,a34a38a139a24a199a154,a12a105a164a139a24a154,a0a143a28a201a89a220a154a139a220a30a88a51a123a77a34,a28a178
a51a24a199a37a222a215a62a24a30a77a34a178a113,a63a215a145a19a30a77a34a178a113,a23a116a123a24a196a88a52a51a24
a199a37a222,a150a149a89a199a178a113a117a166a77a34a123a160a10a139a7a248,a52a202a47a123a178a113a117a123a77a18,a202
a47a123a30a127,a178a113a117a243Euclid a178a193a176a222a52a253a233a123a90a189a234a123,a25a243a89a199a30a127a52a203
a77a34a123a160a10a139a7a248,a89a210a52a201a100,a183a166a98a178x a89a199a37a222da220a39a127a1490,a13a51a248
a123,a183a44a49a233eia89a199a37a222a15d a220a39,a253d(dei) = 0,a79a36a174a241a117d(dei) = 0,a210
a139:
dωij = ωik ∧ωkj,(4.15)
a34a19a222,a105a59a92a143,a183a233dei = ωijej a70a135a73,a108a128a79a143ωija52a24a39a135a73a42,a196a49
a97a148a139a199a139a82,a253a139a123ωij ∧dej,a149a52a139
d(dei) = dωijej?ωij ∧dej = 0,(4.16)
a98a52dej = ωjkek,a196a49a62a24a199dωijeja149a143dωikek a150a139a7a248,a89a52a79a143a28a233
a149j,ka52a70a90,a44a85a196j,a15a44a85a196k,a209a52a441a1163,a195a196a162a123,a79a36a210a122a116
a218a42(4.15),a89a52a24a199a228a253a123a218a42,a23a248a102a38,a217a34a105a59a92,a131a217a52a2433a145
a123a60a111,a105a59a92,a34a19a222,a141a30a89a229a117a523a49a131a90,a98a52a893a49a52a233a149j a70
a90,a79a143ω a52a39a233a193a123,a196a49a183a162a117a23ωik,i negationslash= k,a221a86a210a117a23a23ω13,a243a5
a193a0a199a218a42a23d omega13 a97a148a52ω1jωj3,a98a52a34a19a222,ja144a21a127a1492,a89a52a79
a143j = 1a30ω11 = 0,j = 3 a30ω33 = 0,a196a49a141a30a23a248a52a89a143a248a131a90,a217a34a210a144
7
a139a24a49,a196a49a28a210a122a116
dω13 = ω12 ∧ω23,(4.17)
a51a248a44a49a122a116
dω23 =?ω12 ∧ω13,(4.18)
a89a74a24a196a210a52a196a162a123Weingartena218a42,a58a158a59a92,a196a49a89a74a218a42a209a52a44a59a92
a123a105a135a73a193a253a122a116a123,Weingartena218a42a105a139a63a135,a28a7a199a203a198a117a123ω1,ω2a123
a218a42a7a118a123a155,a113a92a52a24a199a111a231,a253a213a63a116a89a74a218a42a166a218a42(4.10)a35a145.a89a199
a111a231a52a139a1a91a63a66a123,a25a243a199a126a117a189a196a162a123Betti Transform ation,a89a199a183a88
a251a89a234,Weingartena218a42a203a198a117dωia123a218a42a123a35a145a117a52a139a253a47a123a1a91a63a66a123.
3 a77a193a123a228a253a88a35a42
a222a193a183a255a88a234De1a123Levi-Civita a178a113a117,a152a42a82a89a199a37a222a243e3a89a199a48a53a123
a155,a183a210a122a116ω1 ∧ω13 +ω2 ∧ω23 = 0,a253a218a42(4.1 1),a79a36,ω13,ω23a209a52ω1,ω2
a123a34a117a28a92,a183a44a49a85a196
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(4.19)
a132a89a199a183a122a116a89a199a77a193a123a7a155a123a88a35a42,a24a199a77a193a166a12a24a199a77a193a139a31a143a73
a47? a89a199a73a47a243a149a77a71,a77a71a52a77a193a222a123a60a106,a7a48a128,a24a196a123a155,a183a162a126a145
a24a228a253a42a203a145a19a228a253a42a117a44a43:
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (4.20)
a145a19a228a253a42 II = (?dx,de3) = aω2 + 2bω1ω2 + cω22,(4.21)
a233a149a145a24a228a253a42,a183a119a199(I).a199a210a52a77a193a123ds2 = (dx,dx),a132a149dx =
ω1e1 + omega2e2,a98a52e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49a210a122a116ω21 + ω22.
a79a36ωa895a199a24a39a135a73a42a209a139a173a7a123a1a91a63a66,ω1,ω2 a123a178a48a90a210a52a77a193a123
a221a222,a2433a145a56a45a176a62,a77a193a104a108a139a24a199a221a222,a0a143a199a123a221a222a52a31a143a17? a210
a52a199a123Riemann a221a222,a52a24a1992a39a123a135a73a42,a89a1992a39a135a73a42a59a92a244a234,a210
8
a52ω21 + ω22,a0a143,a183a15a44a49a233e3a89a199a37a222a82?(de3,dx),a76a183a166a98a196a85a123,a199
a210a127a149aω2 + 2bω1ω2 + cω22,a89a15a52a24a1992a39a135a73a42,a89a199a24a196a119a41a145a19a228a253
a42,a0a143a139a24a199a145a24a228a253a42,a164a139a199a145a19a228a253a42,a28a210a82a199a123a1a110a138,a1a110
a138a123a90a210a52a + c,a1a110a138a123a232a210a52ac? b2,a24a196a143,H = a+c2 a119a41a77a193a123a165
a77a71,K = ac? b2 a119a41a77a193a123Gaussa77a71,a89a176a7a155a244a234,Gaussa77a71a139a142
a245a139a85a123a117a159,a79a36a89a52a14a143a248a445a199a24a39a135a73a42a132a199a162a123a34a117a28a92a123a7a248
a210a122a116a77a193a123a88a35a42,a165a77a71a203Gaussa77a71a89a220a199a88a35a42a195a85a77a193a123a1a91
a117a159,a7a48a128,a44a49a121a210Gauss a77a71a52a116a123a155,a77a193a52a243a123,Gaussa77a71a52a139
a123a155,a77a193a210a139a145a154,a210a54a106a145a154,a142a245a1a91a117a159a209a44a49a126a89a220a199a77a71a117a195
a85,a196a49a52a77a193a176a62a220a199a33a204a7a123a88a35a42,a89a199a88a35a42a52a60a106,a49a128a183a162
a122a116a123a52a24a39a135a73a42,a13a24a39a135a73a42a88a76a142a52a46a54a196a190a52a31a143a63a135,a98a132
a199a123a228a174a44a49a122a241a60a106a117,a89a60a106a104a108a52a234a137a122a7a118a56a249,a34a122a116a165a77a71
a203Gauss a77a71,a165a77a71a127a1490,a24a196a119a41a244a66a77a193,a196a162a123a244a66a77a193,a222a190a210
a52a28a178a24a30a85a20a123a77a34a56a243a61a6a121a176a62a196a196a123a77a193a210a52a193a232a33a66a123a77a193,
a199a123a165a77a71a1430,a196a49a89a139a59a92a123a1a91a63a66,a33a163,a4a150a25a243a209a139a105a245a7a149a244
a66a77a193a123a207a196,Gaussa77a71a205a7a155,Gaussa77a71a52a127a149ω13 ∧ ω23,a243a77a193a222a15
a24a248a139Gaussa110a243,a77a193a154a154a139a24a199a92a160a27a37a222,a178a89a199a92a160a27a37a222a23a143a24
a199a204a187a1431 a123a69a193a123a154,a210a178a77a193a110a243a116a69a193a222a86a234,a154a199a154a139a24a199a92a160a27
a37a222,a28a2430a154a152a24a199a92a160a27a37a222a203a199a178a113,a199a123a224a154a210a243a92a160a69a193a222,a0a143
a233a149a196a139a123a154a209a41a89a199a232a5a123a155,a210a243a92a160a69a193a222a122a116a24a199a75a173,a243a89a199a110
a243a5,a220a199a193a232a195a163a123a7,a253a199a123a52(imag e)a123a193a232a195a163a203a198a117a193a232a195a163a123
a7a210a52Gauss a77a71,a89a248a24a5a6a210a23a241a117a234,a196a49a89a199Gaussa77a71a139a105a59a92
a123a1a91a63a66,a89a30,a77a193a123a255a88a52a24a199a77a18,a77a193a123a30a127a139a77a34,a77a34a15a139a24
a199Gaussa110a243,a0a199Gaussa110a243a52a82a92a160a35a37a222,a217a34a15a44a49a82a92a160a27a37a222.
a0a143a77a34a123a30a127,Gaussa110a243a178a35a34a110a243a116a92a160a203a222a62,a178a92a160a203a123a221a222a250
a198a117a89a199a221a222a248a210a52a77a71,a25a243a210a178a89a199a10a39a77a18a116a176a145,a77a18a1162a145,a253a77
a18a116a77a193,a196a49a183a132a89a199a77a193Gaussa110a243a116a24a199a92a160a69a193a222a62,a89a220a199a193a232
a123a7a210a52Gaussa77a71,a196a49a89a52a24a199a58a158a7a108a123a1a91a221a222,a25a243a139a199a105a7a155
a123a7a248,a222a62a139dωik = ωij ∧ ωjk,a25a243a183a178a89a199a218a42a126a116ω12,i,k = 1,2,a149
9
a52dω12 = ω13 ∧ ω32,a98a52ωa52a39a233a193a123,a196a49ω32 =?ω23,a89a176ω13 ∧ ω23 a210
a52ac?b2,a15a210a52Gaussa77a71,a196a49a183a210a122a116a218a42
dω12 =?Kω1 ∧ω2,(4.22)
a217a165,Ka52Gaussa77a71,a89a52a24a199a13Gauss a58a158a175a199a123a218a42,a196a49Gaussa178a89
a199a189a174a119a41Theorem Egregious,Egregiousa52a110a182a9a241,a63a135a141a89a52a24a199a234
a88a122a123a189a174,a143a31a143a17? a199a121a210a234Gaussa77a71a144a203a77a193a123Riemann a221a222a139
a7,a203a89a199a77a193a243a56a45a123a160a152a195a7,a79a143a243a89a199a218a42a176a62,ω12a46a178a121a210
a234a144a203a77a193a123Riemanna221a222a139a7,a108a128ω1 ∧ ω2a210a52a243a89a199a221a222a5a123a193a232a195
a163,a104a108a144a203ds2a139a7,a253a144a203Riemanna221a222a139a7,a196a49a175a108Gaussa77a71a52a24
a199a77a193a243a56a45a176a62a123a24a199a88a35a42,a98a52a199a144a203a77a193a123Riemanna221a222a139a7.
a166a233a155a128,a28a178a77a193a35a166(deform)a24a5a6,a38a122Riemanna221a222a88a35,Gaussa77
a71a210a88a35,a196a49Gaussa77a71a139a89a143a173a7a123a117a159,a183a46Gau ssa41a234a142a245a7a155
a123a90a164a224a123a136a42,a89a199a189a174a23a108a52a198a1a47a97a220a123a24a199a136a42,a52a105a1a47a123
a24a199a136a42,a34a19a222,a183a162a210a178a89a74a48a199a126a105a135a73a135a73a24a5a6,a122a116a228a253a218
a42,a108a128a210a122a116a89a74a218a42,a5a39,a183a7a89a132a89a199a218a42a121a210Gauss-Bonneta218
a42,Gauss-Bonneta218a42a52a77a18a174a110a111a174a110a131a90a127a149180o,a82a89a199a218a42a77a18a116
a77a193a123a60a111,Gauss-Bonneta218a42a123a97a126a58a158a131a18,a121a210a123a30a127,a183a41a170a188
a126a203a45,a89a203a45a227a135a126a122a133a236a24a74,a34a189a89a199a45a52a24a199a133a123a34a45(complex
line),a0a143a89a199a133a34a45a123a14a145a52a30a133a34,a243a133a34a176a62,a253a233a138a127a1491a123a133a106
a52a24a199a203a177,a210a52a203a45,a89a199a10a39a243a212a174a222a52a228a253a123,a25a243a76a27a180a122a105a245
a123a52a98a1a91( symplectic geometry),a92a139a98a136a232(symplectic structure)a88a212
a139a126,a28a7a178a98a1a91a126a116a222a6a197a166a123a155,a137a7a28a222a24a199a133a34a45,a210a52a183a162
a25a243a196a89a123a192a220a123a24a199a77a18,a183a162a25a243a892a145,a24a116a212a174a123a155,a56a45a166a30a45
a28a243a24a229a524a145,a196a49a28a228a253a123a56a45a524a145,a2434 a145a56a45a104a165a139a24a199a133a34a45.
a79a36a210a122a116a89a199Gauss a77a71,a25a243a89a199a77a71a522a39a135a73a42,a199da24a5a6a127a1490,
a210a122a116Maxwella48a199,a196a49a135a232a73a243a1a91a222a97a126a52a142a245a212a174a123a228a250,a33a163
a183a243a21a41a166a22a236a147a222a85a234a24a159a169a57,a119a41a21Gauss-Bonneta218a42a166Maxwella48
a199a22,a28a162a152a42a21a73a116,a44a49a86a23a24a23,a183a46a178a89a159a169a57a252a117a201a76a27,a98a52
10
a25a243a79a122a105,a150a73a116,a5a39a183a46a89Gauss-Bonneta218a42a123a24a199a121a210,a89a199a203a183
a162a123a211a42a139a58a158a183a35a123a7a248,a2435 0a35a3,a243a102a210a220a8a201a76,a183a115a135a73a1a91,
a210a122a116Gauss-Bonneta218a42a123a89a199a121a210,a202a47a86a222a133a236a245a234,Gauss-Bonneta218
a42a204a7a63a135a52a128,a28a139a127a91a24a199a77a193,a77a193a222a62a139a24a199a245a30a111,a178a89a199a245
a30a111a123Eulera1a110a106a44a43a171a171a77a71a123a28a92,a89a52Gauss-Bonneta218a42,a183a122a116
a89a199a121a210,a183a46a89a52Gauss-Bonneta218a42a33a59a92a90a33a7a108a123a24a199a121a210,a210a52
a178dω12a24a135a73a210a122a116Gauss-Bonneta218a42a232a73a123a42a6,a108a128a183a243a24a202a143a1a35
a123a30a127,a116a234Princeton a41a176a145a123,a104a108a7a108a13a108a143,a183a46a27a6a178a89a199a77a18
a116a176a145,a89a199a52a44a49a41a116a123,a49a128,a183a164a41a234a74a47a123a211a42,a196a49a44a49a128a183a89
a199Gauss-Bonneta218a42a123a121a210a243
a183a196a41a123a211a42a165a52a183a33a245a161a123a24a199,a183a46a25a243a44a49a136a101a234.
11
a145a202a89 a77a193a88(a19)
a9Gauss-Bonneta218a42
2001a3511a21923a134
1 a77a193a88a21a48a123a59a144
a105a176a108a145a166a76a27a68a193a234,a183a243a26a211a176a212a234a1a21,a28a162a44a49a23a241a117a183a164a150
a139a113a92a80,a88a44a183a250a122a183a164a52a203a76a27a89a89a89a74a192a220,a0a143,a183a152a21a7a89a123
a52Gauss-Bonneta218a42,a89a199a218a42a139a35a104a123a63a66,a15a139a35a104a123a187a36,a131a217a203a183
a199a124a123a211a42a15a139a7a248,a196a49a183a7a10a24a10a183a203a89a199a175a11a52a14a143a248a123a7a248,a183
a162a222a39a89a116a77a193a88,a77a193a88a52a135a73a1a91a176a62a33a173a7a123a24a92a73,a79a143a142a245a135
a73a1a91a123a25a54a243a2433a145a56a45a176a1232a145a77a193a123a60a89a46a178a151a9a234,a51a30,a79a143a199
a52a2433a145a56a45a176a62,a89a199a1a91a123a60a89a52a44a49a23a68a123,a88a52a113a92a126a83a106a117a44a43.
a77a193a88a139a105a159a123a187a36,a33a0a123a104a108a52Monge,Monge a52a27a41a123a76a106a166a27,a198
a144a11a9a233a117a163a139a74a217a196,a196a49a198a248a234a41a76a166a115a71a131a105,a198a233a27a41a123a115a184a139
a105a245a107a47,a198a52a252a187a85a142a5a123a24a199a252a202,a208a207a252a187a85a41a47,a198a52a252a187a85a117a127
a123a48a1a92a159,a4a150a164a203a248a252a187a85a86a132a249a75a67,a79a143a198a123a107a47,a27a41a123a176a211a166
a68(Ploytechnique)a210a79a193a229a117a234,a105a159a24a227a30a207,a27a41a33a80a123a166a9a209a243a176a211
a166a68,a183a46a176a211a166a68a15a142a54a25a243a123a56a149,a139a142a245a80a123a166a9,a190a152a128,a27a41a24a199
a33a76a123a106a166a27Poincare a210a52Ploytechniquea123a166a9,Mongea52a145a24a199a85a7a149
a135a73a1a91a86a123a124,a198a123a86a210a119a41a21a135a232a73a243a1a91a222a123a97a126a22,a89a15a52a183a7a89
a123a11a248,a79a36,a27a41a123a115a184a243a135a73a1a91a139a24a199a35a104a123a4a58,a248a234Mongea253a124
a131a105,a132a149a198a243a176a211a123a107a47,a198a139a105a245a166a9,a209a52a243a135a73a1a91a139a35a104a224a26a123
a124,a108a128a243a135a73a1a91a197a76a21a48a131a165,a7a118a116a24a74a123a52a27a41a12a105a24a199a76a106a166
a27Darboux,Darbo uxa52a27a41a41a166a211a123a179a86a159,a196a49a243a198a123a30a207,a198a52a243a41
a5a142a139a105a245a107a47a123a24a199a124,a198a88a157a52a179a86a159,a15a52a174a169a76a166a174a166a211a123a211a159.
a198a123a33a76a123a211a42a52a143a253a21a77a193a88a22,a183a46,a89a143a76a253a52a106a166a169a26a176a62a123a207a123
a24a199a169a26,a25a243a44a234a132a149a199a52a27a169a123,a105a245a124a88a23a89a199a86,a183a46a89a74a124a233a135
1
a73a1a91a98a232a24a154a234a137,a198a162a97a148a23a89a199a86,Darbouxa123a86a89a122a58a158a80,a221a105a234
a105a245a97a238,a2431941 a35,a183a243a220a8a201a76a115a86,a115a21a135a73a1a91a22,a15a89a116a77a193a88.
a89a116a77a193a88a30,a104a108a210a23Darbouxa123a86,a210a46a116Darbouxa123a86a176a62,a24a199a204a7
a123a48a27a52a126a217a196a41a37,a15a210a52a102a126a217a196a41a37a27,a198a126a122a58a158a180a142,a41a122a58a158
a131a164a224,Darbouxa227a135a88a126a123a24a154a52a198a88a126a105a135a73,a183a46,a183a123a49a52a89a135a232
a73,a13a135a232a73a28a7a89a116a245a195,a245a35a106a123a30a127,a89a199a105a135a73a88a21a27a189,a89a52a79a143
a243a245a35a106a123a30a127,a33a139a72a123a211a228a52a105a135a73,a105a135a73a44a49a28,a62,a44a49a198,a44a49
a135a73,a196a49a139a105a245a83a106a123a228a174a44a49a126a116a105a135a73,a51a30,a24a199a105a135a73a15a52a24a199
a42a6,a89a199a42a6a201a163a105a245a106a166a175a11,a88a11a52a199a123a1a91,a164a52a199a123a73a219,a209a201a28
a105a245a97a238,a79a36a52a58a158a139a126a123,Darbouxa123a98a154a52a198a150a139a126a105a135a73,a198a126a217
a196a41a37a27,a98a52a150a139a105a135a73,a79a36,a139a105a245a211
a42,a88a126a105a135a73,a14a143a205a17? a198a15a52a164a7a126a135a232a73a123,a88a126a105a135a73,a198
a126a160a135a73,a126a160a135a73a7a105a135a73a139a122a245a234,a79a143a28a123a77a193a522a145a123a56a45,a196a49
a233a149a220a199a35a106,a253a77a193a123a107a106u,v,a28a7a233ua70a160a135a73,a233va70a160a135a73,a89a176a62
a139a105a245a160a135a73,a13a126a105a135a73a210a59a92a245a234,a98a52a198a88a102a126a105a135a73,a89a52a105a219
a5a123a47a60,Darbouxa52a21a25a24a39a105a135a73a42a123a145a24a124,a24a199a52Darboux,a24a199
a52Frobenius,a198a162a220a199a124a33a0a21a25a89a199a192a220a123,a98a52a127a116a97a126a116a77a193a207a196a123
a30a127,a88a127a119a143a31a143,a198a150a139a126,a15a142a52a132a149a4a58a123a7a248,a198a85a234a105a135a73a131
a128,a120a209a88a195a234,a196a49a198a88a126a105a135a73a234,a183a166a98a89a234,a2431941a35,a183a166a28a243a102
a210a115a89a199a49,a183a105a7a108a143a46,a143a31a143a88a126a105a135a73a17?a196a49a183a210a126a105a135a73a46a27
a6a41Darbouxa196a41a123a211a42,a221a86a128a150a232a41a77a193a88a90a24a74a1a91a123a255a88,a183a102a126
a105a135a73,a79a36a183a122a116a24a199a105a80a123a234a137.
2 a77a193a88a228a253a19a142a123a195a250
a31a143a119a105a135a73a17?a210a52a28a21a25a7a207a196a77a193a123a155,a77a193a52a24a1992a145a123a22a111,a199a243
a202a47a56a45a176a62a522a145a123,a196a49a199a222a62a127a63a24a199a154a52a220a199a35a106,a47a158a210a119a41a107
a106,a98a52a25a243a17,a183a162a210a119a199a219a92a43a41u,v,a79a36a199a123a43a41a522a199a35a106a123a60a106,
a196a49a52a89a199a30a71a38a199a243a154a24a154a139a24a199a35a178a193,a89a199a35a178a193a104a108a105a7a155,a79a143
2
a183a162a150a139a27a6a207a196a133a236a123a67a111,a183a162a144a21a207a196a33a59a92a123a152a134a34,a178a193a89a74a192
a220,a35a178a193a203a77a193a139a33a183a35a123a7a248,a0a143,a92a128a183a35a123a7a248a88a234,a24a189a7a137a219
a143a21a234a137a251a7a118a205a253a47a123a24a74a175a11,a139a24a199a35a178a193,a243a89a199a35a178a193a222a522 a145
a123,a149a52a154a24a154a210a139a142a245a37a222,a15a210a52a35a37a222,a35a37a222a210a52a203a89a199a77a193a35a35
a123a37a222,a79a143a89a199a77a193a52a243Euclid a56a45a176a62,a196a49a183a44a49a89a89a199a37a222a123a159a221.
a143a59a92a229a68,a183a33a149a255a88a159a221a127a1491 a123a37a222,a253a92a160a37a222,a196a49a139a24a87a92a160
a35a37a222,a203a89a74a92a160a35a37a222a18a134a123a139a12a105a24a199a37a222,a183a162a34a189a199a52a82a196a92a160
a123,a0a143a89a199a37a222a183a162a119a41a92a160a27a37a222,a7a213a63a123a52a89a176a210a139a24a199a1a91a25a54
a21a9a234,a79a143a34a38a89a199a77a193a65a178a234a123a155,a92a160a27a37a222a44a49a53a222a18,a15a44a49a53
a5a18,a166a233a155a128,a89a199a77a193a248a234a52a24a1992a145a123a22a111a131a105,a199a164a139a24a199a189a53,a243
a77a193a222a28a52a126a48a53a221,a164a52a203a31a48a53a221,a89a199a221a196a52a105a88a24a248a123,a196a49,a28a7
a189a14a143a248a6a221a196a52a126a48a53,a89a210a52a7a201a77a193a24a199a189a53,a189a53a139a234a131a128,a199a123
a92a160a27a37a222a210a189a234,a92a160a27a37a222a243a89a199a48a53a44a49a53a222a18,a15a44a49a53a5a18,a189
a234a24a199a131a128,a89a199a77a193a15a210a189a53a234,a89a52a105a173a7a123a24a199a10a39,a175a108a35a139a123a144
a52a24a199a110a82,a98a52a89a52a24a199a105a173a7a123a10a39,Mobiusa52a121a41a149a76a123a1a91a166a27.
a79a143a28a7a189a53,Mobiusa21a25a139a74a77a193a88a21a189a53,a89a104a108a52a105a139a63a135a123a24a71a47
a60,a28a162a76a27a209a127a119a123a89a199a67a111,a210a52a252a24a60a146,a28a178a199a221a24a87a203a229a117a123a155,
a210a122a116a196a162a123Mobius a77a193,a199a150a27a6a189a53,a89a52a1a91a222a105a139a63a135a123a24a71a47
a60,a0a143a183a162a34a189a77a193a46a178a104a189a234a24a199a48a53,a139a234a89a248a189a53a123a77a193a131a128,a1
a91a60a89a52a14a143a248a123a17?a132a92a160a27a37a222e3,a217a165e3 a52a2433a145a56a45,a28a21a25a139a24a71
a47a34,a210a52a128a28a92a213a255a88a77a193a88a234,a28a24a189a7a188a126a77a193a222a123a92a160a35a37a222,a183
a119a89a199a92a160a35a37a222a143e1,a89a248a183a210a139a234a24a199a41a37,a199a139a234a145a24a199a92a160a37a222a90
a145a174a199a92a160a37a222,a152a42a56a45a52a189a53a123,a145a19a199a92a160a37a222e2 = e3 ×e1a15a210a113a92
a104a189a234,a196a49a183a210a139a199a92a160a41a37,a92a160a41a37a210a52a174a199a92a160a37a222a149a76a24a189a123
a39a147,a52a144a35a18a134a123,a143a31a143a92a160a41a37a243a1a91a123a207a196a131a165a52a89a143a173a7? a210a52
a79a143a1a91a52a202a226a228a196a107a207a196a56a45a243a228a196a131a5a88a35a123a1a91a117a159,a13a89a228a196a107
a210a52a41a37a196a196a123a56a45,a79a143a52a139a24a199a63a37a144a139a24a199a228a196a178a24a199a41a37a35a143a217
a199a123a41a37,a150a149a92a14a123a92a160a41a37a203a89a199a228a196a107a123a195a163a196a24a24a233a97,a88a98a52a24
a24a233a97,a13a37a233a97a226a209a95a192a90a24a35a123a117a159,a196a49a228a196a107a105a7a155,a79a143a56a89a123
3
a228a196a88a127a119a243a137a219a123a60a89a131a5a152a91a44a49a255a174,a13a234a41a37a131a128,a210a44a49a255a174a234.
a41a37a210a52a37a222a234,a13a37a222a24a196a52a1393a199a73a222a123a37a222,a13a154a24a199a73a222a52a60a106,a210
a44a49a178a199a135a73,a28,a62a31a143a123,a37a222a139a28,a62a123a228a174,a15a139a135a73a123a228a174,a243a236
a171a63a66a5,a164a44a49a139a232a73a123a228a174,a196a49a183a25a243a210a44a49a135a73,a183a207a196a77a193a123
a30a127,a88a144a24a199a41a37,a0a143a243a77a193a123a154a154,a89a248a123a41a37a139a245a232a17? a34a38a28a65
a122e1a123a155,a51a30a89a199a77a193a52a189a53a123,a89a199a41a37a210a113a92a189a234,e1a52a31a143a17? e1a52
a89a199a77a193a243a89a24a154a123a92a160a35a37a222,a0a143a89a199a77a193a139a245a232a92a160a35a37a222a17? a154a154
a139a24a87a243a35a178a193a222a62a127a149a92a160a37a222,a13a77a193a522a145a123,a196a49a199a162a196a196a123a56a45
a523a145a22a111,a89a52a79a143a89a199a154a52a243a77a193a222a35a196,a522a145a123,a25a243a243a154a189a234a131a128,
a92a160a35a37a222a44a49a119a248a199a221a24a87,a196a24a199a203a177,a196a49a199a52a145a28a24a145,a523a145,a89
a1993a145a56a45a58a158a7a155,a183a46a25a243a34a19a222,a28a162a7a234a137a135a232a73a221a86a234a137a203a135a232
a73a5a86a123a106a166a221a86a243a106a166a165a123a97a126,a89a199a60a89a52a33a59a92a123,a51a30a52a33a139a126a123.
a196a49a183a139a24a1993a145a56a45,a132a149a154a24a154a139a199a203a177,a25a243a139a199a214a9a119a41a203a45,a221a86
a203a177a45,a45a52bundle,a196a49a28a7a207a196a77a193a123a1a91a117a159,a126a89a199a137a219a123a48a27,a24
a189a7a255a88a199a123a203a45,a255a88a203a45a234a131a128,a24a35a209a59a92a234,a79a143a24a35a209a52a37a222,
a13a52a37a222a123a155,a199a139a73a222,a210a44a49a135a73,a210a44a49a126a83a106a221a86a135a73a123a228a174,a183
a162a52a243a255a88a135a232a73,a183a162a34a189a141a116a31a143a60a106a209a44a49a135a73,a183a119a243a89a199a77a193a222
a123a154a143x,a0a143dx a52a24a199a37a222,a210a52a44a198a154a203a248a89a199a154a123a37a222,x a52u,v a123a60
a106,a13u,va52a77a193a222a123a219a92a43a41,a196a49a28a44a49a85a241dx,a34a38x a33a155a243a77a193a222,
a0a143dxa24a189a52e1 a166e2a123a34a117a28a92,a196a49a243a89a199a143a48,a183a210a223a73a188a126a105a135a73a123
a10a39,a34a19a222dxa52a24a199a37a222a138a123a24a39a135a73a42,a196a49a199a52e1 a166e2 a123a34a117a28a92,
a199a123a28a92a248a106a52a24a39a135a73a42,a196a49dxa44a49a85a143
dx = ω1e1 + ω2e2,(5.1)
(dx,dx) a210a52a183a162a77a193a123a169a121a221a222,a79a143e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49
ω21 + ω22 = ds2 (5.2)
a210a52a169a121a221a222,a152a42a89a199a56a249a234,a89a233a149a202a47a89a135a73a1a91a59a92a245a234,a79
a143a202a47a135a73a1a91,a169a121a221a222a7a85a196gijdxidxj,a89a52a79a143a243a35a56a45a176a196a188a126
4
a123a43a41a52a127a63a123Ca rtesiana43a41,a199a88a24a189a18a134,a15a88a24a189a52a92a160,dx a127
a149ω1e1 + ω2e2,a98a52a183a162a105a135a73a139a199a228a253a123a117a159,a210a52a242a126a24a39a123a155,a199a127
a1490,a89a210a52a202a47a128a123a160a135a73a44a49a52a98a166a123a30a71,a24a248a123,a15a210a52a122a116a123a160a135
a73a166a135a73a123a39a147a195a7,a196a49a28a210a178da126a116dxa222a62,a24a189a127a1490,a28a178a141a30a48a13
a123a155,a210a122a116d(ω1e1) + d(ω2e2),a213a63a104a105a135a73a3a193a139a24a199a24a39a79a42a123a155,a135
a73a145a19a199a79a6a7a149a82,a15a13a211a131,a44a49a122a116
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(5.3)
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(5.4)
a183a204a243a5a193a201a241ω12,ω13,ω23,a17a108a126a135a232a73a234,a196a49a44a49a178(xe1e2,e3)a123a135
a73a44a143e1,e2,e3a123a34a117a28a92,a89a199a34a117a28a92a178deia85a196ωijej,a25a243a183a126a135a73a1
a91a202a47a123a110a82,a34a38a139a24a199a141a106a7a173a133a123a155,a210a44a43a35a28,i,j,ka441a1163,a28
a178dei a85a196ωijej,ωija123a1a91a63a66a105a210a23,a28a25a243a139a24a28a41a37,a89a28a41a37a203a24
a28a107a106a139a7a248,a13a233a149a89a24a28a41a37,a210a139a24a199a249a163a41a37,a89a199a249a163a41a37a203a198
a117a41a37a123a7a248a210a52ωij,a89a7a248a52a132a24a39a135a73a42a117a44a43a123,a79a36a210a139
dei = ωijej,(5.5)
a89a28a48a199a42a105a7a155,a199a210a44a43a220a199a249a163a41a37a144a35a123a7a248,a243a89a199a60a89a131a5,
a135a73a1a91a203a197a166a88a76a24a248,a197a166a128a128a35a106a52a30a45,a196a49a24a199a41a37a203a248a30a45a243
a35a196,a79a36a28a114a199a41a37a144a139a24a199a35a106,a209a52a30a45ta123a60a106,a25a243a183a162a52a24a199a77
a193,a154a154a139a142a245a41a37,a196a49a183a89a41a37a123a107a106a523,a89a52a79a143a139a35a193a123a219a92a43
a41,a145a139a35a37a222a243a178a193a176a62a35a166a123a43a41,a196a49a183a25a243a89a199a7a35a106a523,a164a79
a143E a89a56a45a523a145a123,a7a35a106a176a234,a196a49a89a52a139a198a79a38a122a105a135a73a139a72,a183a162a46
a82dei a85a196a48a199(5.5),ωija233a149i,ja52a39a233a193a123,a89a52a79a143a183a123a41a37a52a92a160a41a37,
a253a79a143(ei,ej) = δij,a196a49a199a52a39a233a193a123,a79a36ωija34a19a222a105a59a92,a28a178(ωij)a85
a241a117,a199a52a24a199a48a106,a89a199a48a106a52a39a233a193a123,a196a49a243a233a110a34a123ωa127a1490,a217a155
a123a233a248a233a110a34a52a39a233a193a123,a79a36a34a19a222a144a1393a199a24a39a135a73a42,ω12,ω13,ω23.
a183a46a183a222a39a121a210a234ω12a132dω1,dω2a123a48a199(5.3)a113a92a104a189,a89a52a24a199a173a7a123a189
5
a174,a89a52a38a122Levi-Civitaa241a214a123a173a7a189a174,a183a25a243a178a48a199(5.5)a70a105a135a73,a79
a143d(dei) = 0,a196a49a141a30a123a155,a183a210a122a70dωij,a136a42a122a116a123a52
dωij = ωik ∧ωkj,(5.6)
a79a36a89a74ωa131a45a139a105a59a92a123a7a248,a59a92a122a88a122a234,a79a143a31a143a17?a79a143a233
a149dωij = ωik ∧ωkj,i,ja52a88a35a127a123,a152a42a35a127a234a123a155,ωii a520,a89a52a79a143ωa52
a39a233a193a123,a196a49a28a82i negationslash= j,a152a42ka127a149i,a11ωii = 0; ka7a127a149j,ωjj = 0,a196a49k
a88a127a149i,a88a127a149j,a79a143a183a162a52a2433a145a56a45,ka144a139a24a199a44a21a117,a79a36a89a199a23a248
a105a0a219a123a48a199a42,a199a123a141a30a144a139a24a49,a183a222a39a178a199a85a5a117a234,a210a52
dω12 = ω13 ∧ω32,(5.7)
dω13 = ω12 ∧ω23,(5.8)
dω23 = ω21 ∧ω13,(5.9)
a131a217a52a122a116dω12 = ω13 ∧ω32a89a199a218a42,a98a52ω a52a39a233a193a123,a196a49a210a122a116
dω12 =?ω13 ∧ω23,(5.10)
a13ω13,ω23a209a52ω1,ω2a123a34a117a28a92:
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(5.11)
a89a166a28a210a122a116a5a193a89a199a218a42:
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(5.12)
Ka52Gaussa77a71,Gaussa77a71a210a52K = ac?b2,a89a199a218a42a88a122a234,a104a35Gaussa88
a52a89a248a122a116a89a199a218a42,a52a126a107a123a48a27a122a116a218a42,a199a119a41Theorem Egregium,
a126a165a169a89,a199a52a24a199a219a201a123a189a174,a201a123a189a174,a28a251a251a23a23a199,a199a52a105a201a123,a79
a143a183a162a123Ea52a92a160a37a222a45,a199a89a199a203a177a45a233a149a77a193M a139a24a199a61a107,a233a149a203a177
a45,a139a89a199a92a160a37a222,a183a82a199a123a198a154a210a52a199a123a61a107,a79a36a183a162a25a243a123a1a91a7
a44a3a10a39a222a7a118a133a236a234,a210a52a128,a88a144a52a139a24a199a77a193a221a86a139a24a199a77a34,a25a243
6
a139a220a199a56a45,a139Ea89a199a56a45a90a77a193M,a47a34a222,a139a234a77a193M,a108a128a139a132a196a139a199
a123a92a160a37a222a196a196a123a56a45a52E,a199a52a77a193a123a203a177a45,a25a243a47a158a119a89a199a77a193a52a142
a56a45,a199a52a243a142a222a123a24a199a56a45,a196a49a210a139a24a1993a145a123a203a177a45,a199a123a142a56a45a52a183
a162a123a77a193,a243a89a199a60a89a131a5,a183a162a139a1a199a24a39a135a73a42,a243a56a45a176a62a139a24a39a135
a73a42ω1,ω2,a108a128a139ω12,ω13,ω23,a24a2255a199a24a39a135a73a42,a217a165ω13,ω23a52ω1,ω2a123
a34a117a28a92(a218a42(5.11)),a199a44a43a77a193a123a1a91a117a159,a183a162a139
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (5.13)
a145a19a228a253a42 II = (?dx,de3) = ω1 ∧ω13 + ω2 ∧ω23 = aω2 + 2bω1ω2 + cω22.
(5.14)
a15a210a52ω1∧ω13 a28a222ω2∧ω23a52a77a193a196a162a123a145a19a228a253a42,a145a24a228a253a42a52ω21+ω22,
a52a220a199a35a106ω1,ω2a123a178a48a90.a145a19a228a253a42a233a149a145a24a228a253a42a139a1a110a138,a1a110a138a123
a83a106a90a24a196a119a41a165a77a71H = a+c2,a1a110a138a123a232a210a52Gaussa77a71,a196a49Gaussa77
a71a52ac?b2,a13a183a162a25a243a139a24a199a219a201a123a218a42a210a52dω12 =?Kω1 ∧ω2,a243a89
a199a218a42a141a30,ω1 ∧ω2 a210a52a77a193a123a193a232a195a163,a199a104a108a144a203a77a193a123a221a222a139a7,a166
a199a123a160a152a150a139a7a248,a144a203a77a193a123a145a24a228a253a42a139a7,a243a199a123a38a30,ω12 a52a183a162
a123Levi-Civitaa201a100,a132a149a141a30a132a145a24a228a253a42a113a92a104a189,a196a49a38a30a15a52a144a166a145
a24a228a253a42a139a7,a79a36a183a162a127a119Gaussa77a71a144a203a145a24a228a253a42a139a7,a89a52Gaussa58
a158a122a63a123a136a42.
3 Gauss-Bonneta218a42
a183a162a44a49a132dω12 =?Kω1 ∧ ω2 a122a241a117Gaussa139a214a123a89a199a189a174,a183a25a243a128,
Gauss-Bonnet a218a42a15a193a47a210a132a89a199a218a42a122a241a117,a0a143,a104a35a183a15a89a143a46,a79
a143a34a38a28a139a24a199a85a20a123a77a193,a139a234a89a199Gauss a77a71a198a49a193a232a195a163,a28a104a108a105
a7a108a143a175a199a123a232a73a52a31a143,a139a24a199a193a232a195a163a198a49a24a199a60a106(Gaussa77a71),a178
a199a243a77a193a222a232a24a40a122a241a123a106a52a88a52a203a89a199a77a193a123a1a91a139a7?a0a143,a89a199a210
a52Gauss-Bonneta218a42,Ma52a85a20a123a189a53a77a193,a11
integraldisplay integraldisplay
KdA = 12piχ(M),(5.15)
7
a243a183a162a89a199a218a42a165,a183a162a178ω1 ∧ ω2 a119a193a232a195a163,a183a162a15a44a49a119a199a41dA,
KdAa210a52a89a199a232a73,a28a193a47a210a46,a183a162a243a85a20a123a77a193,a196a49a152a42a139a24a199a232
a73,a199a44a49a85a196da31a143a123,a0a143a199a123a232a73a243a97a148a52a243a77a193a123a30a142a222,a44a49a35
a143a24a199a30a142a232a73,a89a52a183a13a41a89a123Stokesa189a174a204a7a123a92a73,a196a49a204a88a204a132
a89a199a122a116Gaussa77a71a123a232a73a520,a89a23a108a52a70a216a123,Gaussa77a71a232a73a88a204a127
a1490,a89a52a79a143a28a82a24a1992a145a123a69a193,a199a44a49a82a24a196a123a221a222a38a199a123Gaussa77a71
a127a1491,a0a143a243a77a193a222a123a232a73a104a108a88a204a127a1490,a196a49a89a199a119a174a52a70a216a123,a104
a35a79a36a183a46a70a243a31a143a143a48,a70a243a183a162a89a199a218a42dω12 =?KdAa88a52a243a77a193a222,
a52a243a203a177a45a222a62,a89a199a77a193a52a243a203a177a45a222,a196a49a183a137a7a178a89a199a77a193a10a176a116
a203a177a45a222a62,a10a176a52a31a143a63a135a17? a210a52a243a154a24a154a28a82a89a199a154a42a143a198a154a123a92
a160a35a37a222,a196a49a166a233a155a128,a183a162a24a196a119a89a199a143a37a222a156,a79a36a28a243a24a1992a145a22
a111a222a193a7a82a24a199a37a222a156,a89a199a37a222a52a92a160a37a222a203a77a193a35a35,a253a143a92a160a37a222
a156,a89a52a88a52a44a21? a177a34a127a201a28a24a199a77a193,a28a21a88a21a154a154a141a189a24a199a92a160a35
a37a222,a221a86a35a37a222,a38a122a199a52a203a153a123,a4a150a52a44a49a135a73a123,a89a248a123a37a222a156a88
a24a189a139,a89a52a24a199a95a192a123a189a174,a114a199a123a89a24a227a123a21a48a52a77a193a88,a221a86a52a114
a14a77a193a88a203a77a193a123a221a222a89a171a171a123a7a248a243a139a88a245100a35a3a52a204a7a123a175a11,a0
a199a30a127a33a204a7a123a236a147a52a21Mathematische Annalena22,a92a45a142a106a166a123a165a101a52
a243a121a41,a21Mathematische Annalena22a222a62a7a149a89a199a175a11a139a105a245a169a57,a0a30
a127,Einsteina52a21Mathematische Annalena22a123a24a199a31a246,a89a199a175a11a113a92a180a56
a249a234a137a7a24a74a30a45,a0a143a217a165a139a24a199a79a163a52a34a38a28a139a24a199a85a20a123a77a193,a28
a82a154a154a123a63a37a189a53a123a92a160a27a37a222,a253a82a154a154a104a189a123a92a160a27a37a222,a178a199a110a243
a116a92a160a69a222a86,a89a199a47a158a119a41Gauss a110a243,a196a49Gaussa110a243a178a28a123a77a193a110a243
a116a92a160a69a222a86,a89a248a210a178a24a1992a145a77a193a110a243a116a12a24a1992a145a77a193,a220a199a209a52a189a53
a123,a149a52a89a176a210a139a24a199a95a192a88a35a42,a119a41Mapping Degree,a253a110a243a123a39a106,a243
a178a193a123a60a111,a28a139a24a30a77a34,a196a49a92a160a69a210a52a92a160a203,a0a143a28a178a89a30a85a20a123
a77a34a110a243a116a92a160a203a222a62,a196a190a199a221a234a245a232a87a17?a89a210a52Mapping Degree(a110a243
a123a39a106),a89a199a189a66a243a176a145,a4a150a522a145a210a88a89a143a59a92a234,a23a108,a183a162a178a77a193
a110a243a116a92a160a69a222a62,a108a128a82a92a160a69a222a62a123a193a232a195a163,a89a210a52Gauss a77a71,a196
a49a89a199a110a243a210a127a149Gaussa77a71a123a232a73,a178a89a199a233a193a56a45a123a193a232a195a163a126a110a243a110
8
a195a117a131a128a90a198a117a77a193a123a193a232a195a163a24a248,a253a143a70a199a123a114a199a123a232a73,a210a122a116a110
a243a123Degree,a44a49a121a210a89a199Degree a243a44a49a189a53a123a77a193a123a60a89a5,a199a52a77a193
a123Eulera106a123 12pi,a105a44a234a123,a183a76a150a150a139a30a45a178a89a199a113a92a89a113,a183a46a183a5a39
a242a45a251a89a89,a183a25a243a76a151a89a89a89a199a121a210a52a14a143a248a123,a23a108a183a162a46a139a228a253a218
a42dω12 =?KdA,a183a162a164a7a178a77a193a56a116Ea222a86,a196a49a7a189a24a199a92a160a37a222a156,
a89a88a24a189a44a21,a89a248a123a92a160a37a222a156a7a139a73a154,a15a210a52a128a152a42a28a227a142a114a14a92a160
a37a222a156a139a73a154a123a155,a199a0a52a44a21a123,a89a15a137a7a121a210,a139a234a73a154a131a128,a210a23a89a199
a73a154a123a117a159,a7a152a128,a28a23a23a89a74a92a160a37a222a156,a152a89a1a199a67a195a67a81a196,a89a52a37
a222a156a123a73a154a123a44a21a117,a44a21a37a222a156a139a142a245a88a51a123a48a53,a98a52a243a73a154a139a30a127a89
a199a37a222a53a105a18,a15a139a30a127a37a222a53a176a18,a15a139a30a127a210a54a145a174a199a67,a199a52a203a24a24
a118a77a34a35a35,a196a49a255a88a37a222a156a123a73a154a123a117a159a52a58a158a139a63a135a123a175a11,a0a143a183
a243a127a91a77a193a24a189a139a24a199,a63a37a37a88a144a24a199,a13a52a139a142a245a139a73a154a123a92a160a37a222a156.
a243a89a199a60a111a131a5,a28a7a178a77a193a35a196a66a76,a35a196a24a199a196a162complex(a133a111),a28a11
a189a199a243a186a154,a79a143a196a162a123complex,a210a52a128,a139a74a186a154,a139a74a30,a139a74a193,a243a186
a154,a189a24a199a127a91a123a37a222,a0a143a145a19a90a178a89a199a37a222a156a210a116a30,a82a24a199a30a123a155,a152
a42a37a222a156a46a178a243a186a154a104a189a234,a0a143a105a142a52a23a241a117a44a49a178a199a210a116a114a199a123a30,a25
a243a52a193a234,a34a38a52a193a123a155,a7a48a128a139a199a174a110a111,a25a243a37a222a156a46a178a243a30a222a209a189
a234,a52a88a52a21a234a210a159a116a199a123a19a92,a89a210a88a210a23a234,a104a108a89a52a24a189a44a49a123,a34
a38a28a227a142a199a139a73a154,a7a48a128a52a24a199a174a110a111,a0a143a28a210a243a174a110a111a165a45a82a24a154,
a108a128a44a89a154a203a116a30a222,a0a143a89a199a37a222a156a210a178a199a210a116a19a92,a98a52a89a248a189a123a37a222
a156a243a165a45a123a0a154a24a189a52a73a154,a79a143a165a45a123a0a154a150a27a6a189a24a199a44a49a203a153a123a37
a222a156,a38a122a154a154a144a139a24a199a37a222,a243a89a24a154a210a150a27a6a189,a196a49a166a98a89a199a174a132a44
a49a121a210,a34a38a28a227a142a89a37a222a156a139a73a154,a89a248a123a37a222a156a52a63a243a123,a202a47a24a189a139
a73a154,a243a143a69a222,a7a253a90a123a155,a89a90a203a143a69a35a35a123a30a127,a199a123a48a53a52a199a37a222
a156,a24a189a139a24a154a150a139a90,a150a232a139a24a154,a13a89a24a154a210a105a133a236a234,a0a143a89a171a248a6a154
a123a207a196a52a105a139a63a135,a15a105a7a155a123a175a11,a75a47a37a222a156a243a73a154a123a117a159,a105a59a92a123
a24a199a48a27a52a183a162a44a49a243a73a154a189a24a199a114a106,a89a24a196a210a119a41a199a123a141a41a213
I(s) = lim 12pi
integraldisplay
ω1,(5.16)
9
a210a52a128,a165a45a139a73a154,a73a154a142a163a123a37a222a156a52a113a92a104a189a123,a196a49a28a243a89a154a152a199
a66a203a177,a0a143a203a177a123a30a142a37a222a156a46a178a139a234,a196a49a28a243a89a24a154a178a199a110a243a116a24a199
a203a177,a79a143a219a92a123a30a127,a28a34a189a52Euclida1a91,a34a189a126a178a113a34,a28a210a178a199a110a116
a203a177,a0a143a119a203a177a52a245a232a39a17? a89a210a52a37a222a156a123a141a41,a5a39a183a121a210a34a38a28a139
a234a89a74,a28a210a82a92a160a37a222a123a37a222a156,a63a37a178a89a199a77a193a10a176a116Ea176a62,a253a10a176
a116a203a177a45a176a62,a199a210a35a89a199a77a193a143a139a43a154a123,a17a139a43a154,a145a139a73a154,a89a199a73a154
a65a222a86a210a52a43a154,a0a143a89a74a43a154a123a141a41a28a243a24a229a210a52Euler a158a106,a210a127a149a89
a199a232a73,a253a89a199a232a73a139a63a66,a199a52Eulera158a106,a183a46a135a232a73a139a24a199a173a7a123a97a126
a52a243a133a35a60a106a88,a28a7a89a106,a33a139a63a135a123a106a210a52a133a106,a105a110a99,a183a23a165a41a106a166
a36,a165a41a124a212a34a19a234,a165a41a124a150a139a133a106,a133a106a7a155a122a88a122a234,a183a7a89a24a154a133
a35a60a106,a183a7a121a83a106a228a253a189a174,a133a35a60a106a131a128,a127a91a83a106a48a199a42a209a139a137,a89a52
a88a122a234a123a24a199a136a42,a89a199a136a42,a104a35Eulera88a204a121,a105a245a124a209a121a88a241a117,Gauss
a21a121a210,a198a52a163a83a33a149a76a123a106a166a27.
10
a145a24a89 a77a193a88(a174)
a9Gauss-Bonneta218a42(a153)
2001a3511a21930a134
1 a135a232a73a243a133a35a60a106a88a165a97a126a59a144
a183a164a97a148a242a89a220a39,a89a220a39a183a139a199a15a153,a189a247a89a24a154a133a35a60a106a88,a79a143a243a106
a166a165,a105a7a155a123a24a71a47a34,a51a30a243a106a166a36a222a15a52a58a158a7a155a123a24a71a47a60,a210a52
a139a133a106,a89a199a133a106a38a122a106a166a59a92,a133a60a106a139a142a245a164a224,a139a63a135a123a117a159,a79a36,
a89a38a122a89a74a60a106a243a97a126a222a1a47a139a126a255,a196a49,a183a189a247a89a24a89,a7a152a128,a133a35a60
a106a139a24a199a105a173a7a123a117a159,a127a63a123a83a106a48a199a243a133a35a60a106a131a165a24a189a139a137,a89a52a24
a199a88a122a234a123a47a60,a79a143a88a11a28a14a143a248a85a24a199a48a199,a28a7a52a227a142a137a52a133a106a123a155,
a199a24a189a139a137,a190a152,x2 + 1 = 0,a0a143a199a139a199a137a210a52√?1,a196a49√?1a210a89a143a248
a6a139a126a255,a88a98a152a36,a133a106a203a34a106a24a248,a44a49a28a62,a139a51a248a123a117a159,a196a49,a199
a44a49a228a174,a51a30a199a221a57a234a142a245a97a238a52a34a106a88a21a221a57a123,a183a46a183a123a49a243a44a199a165
a24a189a204a139a199a56a105,a243a56a105a123a30a127,a183a46a73a220a39a89a133a35a60a106,a183a189a247a89,a24a199a52
a183a166a98a89a123a83a106a123a228a253a189a174,a210a52a128a127a63a123a83a106a123a48a199a243a133a106a173a165a24a189a139
a137,a89a199a52a105a10a121a210a123,a137a7a106a166a222a99a123a10a39,a7a48a128,a149a76a106a166a27a152Euler,
a198a46a27a6a121a210,a98a150a139a21a196a213,a183a46Gauss a52a183a162a163a83a33a149a76a123a106a166a27,a198
a105a35a52a123a30a127a210a139a24a199a121a210,a15a210a52a133a106a137a7a24a74a1a91a123a117a159,a88a113a92a52a83
a106a123a175a11,a183a189a247a5a39a89a133a106a123a30a127a121a210a89a199a189a174;a51a30,a133a35a60a106a33a204a7
a123a24a199a189a174a52Picarda189a174,a210a52a128,a34a38a233a149a24a199a133a35a60a106,a82a199a123a60a106a138a243
a133a178a193a176a62a196a82a123a160a152,a199a178a114a199a133a178a193a209a152a212a234,a217a165a15a142a44a49a86a171a24a154,
a220a154,a89a52a88a122a234a123,a210a52a128,a60a106a152a42a52a24a199a92a24a60a106a123a155,a199a73a89a122a58a158
a131a254a225,a44a49a128a139a88a245a178a178a193a209a152a212a234,a139a63a135a123a24a71a47a60a52a89a199a189a174a52a133
a35a60a106a176a88a123a189a174,a44a49a188a126a183a162a25a243a7a89a123Gauss-Bonneta218a42a117a121a210,a89
a128a210a23a229a117a150a139a7a248a123a24a74a48a27a203a10a39,a136a42a52a139a7a248a123,a89a52a106a166a222a58a158
a7a155,a139a63a135a123a175a11.
1
2 a7a149a166a243a123a7a196a117
a89a199a49a79a136a101a234,a28a162a243a89a199a49a85a199a231a183,a33a80a52a7a196,a28a21a234a7a3a73a116a24
a199a175a11,a89a52a205a7a155a123,a183a46a28a162a209a52a76a166a9,a76a166a9a73a176a127a115a184a33a128a123a24
a227,a49a128a116a246a204a222a86,a253a38a243a166a68,a243a166a98a92a160a176a62,a33a7a155a123a24a189a7a7a196.
a88a7a52a127a144a19a119a28a41a31a143,a28a242a41a31a143,a89a199a33a160,a7a7a196,a7a7a3a21a73a175
a11,a7a7a3a21a234a73a133a7a3a73a123a175a11,a0a143,a104a108a28a73a123a175a11a88a24a189a92a55,a28a246
a30a15a88a24a189a21a234a122a116a73a153,a88a44,a28a165a45a178a44a24a74a107a52,a178a44a24a74a70a216,a44a49
a38a122a28a123a166a175a93a116a159a90,a13a38a122a28a93a116a159a90a123a210a52a7a178a44a89a248a123a67a153,a196a49
a183a243a185a76a27a7a7a196,a245a24a154a143a89a229a117,a28a162a4a150a7a21a234a28a132a24a199a76a14,a144a35
a231a183a73a175a11,a221a86a62a68a19a68a105a144a19,a51a166a117a41a231a183,a89a52a105a139a80a255a123,a7a3a7
a178a106a166a46a24a46,a221a86a233a127a63a123a166a175,a28a7a3a139a199a135a46,a250a122a139a199a31a143a248a123a217
a196,a233a149a28,a233a149a89a199a166a175a123a127a35a44a49a15a28,a51a30a28a233a166a175a123a21a197a15a44a49a15
a28,a196a49a89a52a105a138a122a213a63a123a24a71a47a60,a230a130a28a162a31a68a24a5a89a199a44a21a117.
3 Gauss-Bonneta218a42a123a121a210
a222a39,Gauss-Bonneta218a42a183a150a139a121a210a92,a196a49a183a11a178a121a210a128a92a234,a183a222a39a89
a123Gauss-Bo nneta218a42a210a52,a34a38a243a56a45a176a62a139a24a199a77a193,a199a52a24a199a114a199a123
a77a193,a63a37a34a38a89a199a77a193a52a189a53a123,a253a199a123a27a34a139a24a189a123a48a53,a149a52a89a248a6,
Gaussa77a71Ka210a52a77a193a222a123a24a199a60a106,a183a44a49a178a89a199a60a106a233a149a77a193a222a123a193a232
a221a222a70a232a73,a89a199a232a73a52a24a1992a173a232a73,a70a199a232a73a131a128,a136a42a89a199a232a73a127a149a24
a199a158a106(2pi)a198a49a77a193a123Euler a43a117a106,a253
integraldisplay integraldisplay
KdA = 2piχ(M) (6.1)
Eulera43a117a106a210a52a178a77a193a35a196a66a76a131a128,a55a149a24a154a7a108a123a30a71,a178a199a35a113a131
a128,a217a186a154a199a106?a30a123a199a106+a193a123a199a106,a89a2483a199a106a123a116a139a123a90a210a119a41a89a199a77
a193a123Eulera43a117a106,a104a77a193a52a69a193a123a155,a199a123Eulera43a117a106= 2,a152a42a199a52a199a162
a193,a199a123Eulera43a117a106a520,a28a162a44a49a65a24a65,a210a21a122a116a89a199.a152a42a77a193a52a199a189
2
a53a77a193,a89a52a199a123a141a24a123a95a192a88a35a42,a24a196a89a229a117,a34a38a69a222a28a1a199a162,a162a123
a199a106a210a203Eulera43a117a106a139a199a7a248,a89a162a123a199a106a202a47a119a77a193a123a90a194(genus),a89
a52a77a193a33a173a7a123a95a192a88a35a42,a139a63a135a123a52,a89a77a193a123a117a159,a77a193a222a62a60a106a123
a117a159a203a90a194a139a183a35a123a7a248,a196a49a90a194a52a95a192a88a35a42,a199a107a47a116a77a193a123a1a91a117
a159a90a137a219a117a159,a139a58a158a131a173a7a123a107a47,a196a49a114a199a89a74a7a248a52a105a253a163a123,a35a104
a253a163a123,a79a36,a15a52a58a158a7a155,a58a158a139a63a135a123,a183a222a39a121a210Gauss-Bonneta218
a42,a33a7a155a123a218a42a210a52
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(6.2)
a183a25a243a173a133a24a40,a7a207a196a77a193a88a123a155,a24a189a7a207a196a77a193a222a123a41a37,a34a38a82a89
a199a41a37,a38a41a37a1233a199a92a160a37a222a144a35a18a134,a63a37a183a162a34a189a199a52a199a141a67a248,a253
a243a220a199a131a45a160a10a24a199a141a67a221a86a38a67,a183a162a34a38a52a141a67a248,a0a143,a233a149a89a248a6
a41a37,a34a38a28a127a119a145a24a199a37a222a131a128,a217a199a220a199a37a222a210a104a189a234,a79a143a183a162a34a189
a145a174a199a52a77a193a123a92a160a27a37a222,a0a143a145a24a199,a145a174a199a189a234a123a155,a145a19a199a15a210a189
a234,a47a34a222,a183a89a52a24a199a92a160a41a37,a51a30a52a141a67a248(a141a67a41a37),a89a210a113a92a189a234.
a196a49a233a149a243a24a199a154a123a196a139a89a171a248a6a123a41a37,a24a225a89a171a41a37a139a92a107a106a248(one
parameter family),a52a202a226a234a24a199a35a106,a77a193a522a145a123,a242a28a222a89a154a123a41a37a139
a24a199a107a106,a196a49a77a193a196a139a41a37a52a24a1993a145a123a56a45,3a145a56a45a139xa89a199a186a154,a189a199
a243a77a193a123a160a152,a199a86a171a220a199a145,a108a128a242a82a24a199a35a34a48a53,a145a139a24a199a145,a79a143a35
a34a243a35a193a176a62a44a49a221,a196a49a145a245a234a24a145,a89a248a6a210a122a116a196a139a41a37a248a196a196a56a45
a1233a145a123a117a159,a139a234a41a37a248,a139a31a143a80a255a17? a79a143a139a234a37a222,a28a44a49a126a218a42a117
a44a43a241a117,a37a222a139a73a222,a89a73a222a210a139a106,a183a162a180a106a166a33a7a155a123a7a139a106,a28a7
a139a106a123a155,a195a85a52a239a104a123,a63a37a97a126a123a30a127a28a44a49a10a137a116a123a209a52a106,a243a236a171
a63a66a5a143a31a143a135a232a73a7a155? a183a46a106a166a204a7a123a248a123a52a207a196a60a106,a207a196a220a199a248a58
a123a7a248,a25a243a89a7a248a17,a60a106a88a80a180a234,a196a49a135a232a73a52a178a89a199a7a248a34a117a154,a79
a36a44a49a126a83a106,a37a222a44a49a28,a252a199a106a248a117a198,a196a49a135a232a73a204a7a123a196a210a52a178a56
a45a123a174a88,a178a60a106a123a174a88a34a117a154,a83a106a154,a139a234a83a106a49a128,a28a210a44a49a174,a196a49
a210a139a126,a79a36a15a173a7,a0a143a139a234a41a37a196a122a116a123a137a219a123a47a34a52a31a143a17? a183a178a89
3
a199a41a37a119a41xe1e2e3,
E = {xe1e2e3|Ma189a53 e3a52a27a37a222,e1 mbox4#%} (6.3)
e3a52a92a160a123a27a37a222,x,e1,e2,e3 a209a52a37a222,a196a49a199a162a123a135a73a15a52a24a199a37a222,a135
a73a131a128a52a24a39a135a73a42a37a222a138,a79a36,a199a162a44a49a44a143e1,e2,e3a123a24a199a34a117a28a92.
a183a178dxa44a143a34a117a28a92,a122a116a123a248a106a183a119a41ω1,ω2,
dx = ω1e1 + ω2e2,(6.4)
ω1 ∧ ω2a210a52a77a193a123a193a232a221a222,a52a24a1992a39a135a73a42,a199a104a108a44a49a126a117a41a199a173a232
a73a123a232a73a60a106(integral),a196a49a178a199a232a73a123a155,a210a122a116a89a199a77a193a123a193a232,a183a89
a123a7a149a77a193a123a174a88a123a89a74a136a42,a28a243a135a73a1a91a86a222a73a88a116,a152a42a28a88a21a113a92
a122a73,a88a21a113a92a195a123a155,a150a139a7a248,a79a143a89a74a19a142a76a150a52a202a47a135a73a1a91a44a49
a89a24a199a219,a183a89a24a220a39a210a178a199a89a113a234,a89a15a121a210a89a199a48a27a123a128a154,a199a123a128a154
a204a7a52a183a243a207a1963a145a56a45a123Euc lid a1a91,Euclida1a91a33a80a52a126a116a98a41a37,a79
a143a116a98a117a243Euclida1a91a88a35,a52a139a63a66a123,a196a49a33a80a126a116a98a41a37,a0a143,a24a196
a123a135a73a1a91a123a86a127a126a116a77a193a88a123a30a127,a199a88a126a116a98a41a37,a28a7a46a47a123a27a6.
a7a152a128,a178a193a137a219a1a91,a28a88a126a116a98a41a37,a28a123a220a199a43a41a88a18a134,a4a150a149a199a18
a123a48a53a88a52a92a160a123a48a53,a28a86a65a65a23a210a127a119a10a245a234,a102a38a245a234,a88a52a88a44a21
a41,a44a49a41a116,a210a52a102a38a245a234,a139a63a135a123a24a71a47a60,a104a108a183a162a209a127a119,a43a41a248
a58a52a27a41a123a83a166a27,a106a166a27a136a11a16a21a25a123,a198a62a24a39a126a43a41a123a30a127a88a52a116a98
a41a37,a209a52a127a63a123a41a37,a198a126a127a63a123a41a37a252a117a255a174a89a171a1a91a123a175a11,a88a127a119
a52a253a160a11a9a56a234a199a116a98a41a37,a49a128a28a243a86a222a23a116a123a209a52a116a98a41a37,a196a49,a183
a123a41a37a52xe1e2e3,a894a199a209a52a37a222,a199a162a123a135a73a15a122a116a37a222a138a123a24a39a135a73,a196
a49a154a24a199a44a49a44a143e1,e2,e3a123a34a117a28a92,a132a149a183a162a52a243a24a1993a145a123a56a45,a0a143
a89a210a52a222a193a85a123a89a199a218a42(6.4)a90a5a193a123a218a42:
dei = ωijej,(6.5)
a89a30a127,a79a143a52a24a39a135a73a42,a196a49a89a171a34a117a28a92a52e1,e2,e3a123a34a117a28a92,a199a123
a248a106a52a24a39a135a73a42,a88a242a52a60a106a234,a49a3a152a42a52a60a106a123a155,a199a34a117a28a92a123
4
a248a106a52a60a106,a25a243,a248a106a52a24a39a135a73a42,a89a74a24a39a135a73a42a173a7a122a105,a79a143a199
a195a85a24a199a41a37a203a199a248a163a41a37a123a7a248,a199a248a163a41a37a196a24a154a154,a203a198a117a35a139a245
a232? a35a139a52a24a199a135a73,a210a52a183a162a123ωia203ωij,a89a1a199a135a73a42a139a59a92a123a7a248,a33
a7a155a123a52ωij,a28a23a199a105a102a38,i,ja441a1163,a98a52a79a143a41a37a52a116a98a123a92a160a37a222,
a196a49ωija233a149i,ja52a39a233a193a123,a79a36,a28a178ωija85a196a24a1993 × 3 a123a48a106a123a155,a89a199
a48a106a52a39a233a193a123,a199a123a233a110a34a123a195a163a209a520,a63a37a233a149a233a110a34a199a52a39a233a193a123,
a196a49a144a1393a199a93a116a7a255a174a123a24a39a135a73a42,a28a7a126a41a37a117a207a196a1a91a123a89a171a60
a89,a243a197a166a105a7a108,a197a166a89a24a199a212a14a243a0a14a35a196,a0a143a199a123a160a152a210a52a30a45a123
a60a106,a79a36,a89a41a37a52a30a45a123a60a106,a89a171a60a106a243a197a166a222a52a24a199a35a106a123a60a106.
a79a143a243a197a166a222,a243a196a197a166a222,a93a116a123a35a106a52a30a45,a144a139a24a199,a98a52a7a207a196a1
a91a123a155,a60a89a117a122a133a236,a44a21a89a199a41a37a52a203a245a149a24a199a35a106a139a7a248,a44a49a52a245
a35a106a123a60a106,a79a36a89a131a45a210a139a74a7a248,a89a7a248a210a52a28a70d(dei),a183a89a44,a28a126
a222da123a155,da126a220a39a520,a196a49a28a178a89a199a7a248a85a241a117a123a155,a210a122a116dωij a52a24a199a42
a6,a44a49a126a217a198a123ωa117a44a43,a89a42a6a52
dωij = ωik ∧ωkj,(6.6)
a28a122a116a89a248a6a24a28a48a199,a89a52a139a63a66a123,a79a143ωija52a24a39a135a73a42,a28a178a199a135
a73a123a155a522a39a135a73a42,a13a243a141a30a52a220a199a24a39a135a73a42a35a198,a196a49a15a522a39a135a73
a42,a79a36a89a28a48a199a88a175a216,a89a28a48a199a58a158a7a155,a199a162a83a44a228a196a107a114a199a123a117
a159,a89a28a48a199a23a248a133a236,a217a34a58a158a59a92,a79a143a89a74ωij a52a39a233a193a123,a196a49a152
a42i negationslash= j a123a155,a190a152,a152a42i = 1,j = 2,a0a143k = 3,a89a52a79a143k a7a52a127a1491,a149
a52a139ω11 = 0,a13a7a52ka127a1492,a0a143a139ω22 = 0,a196a49a89a28a48a199a42a23a248a133a236,a141
a30a144a139a24a49,a105a59a92a143,a28a164a44a49a122a116a24a199a1a47a60a111,a210a122a116
dω12 = ω13 ∧ω32 =?ω13 ∧ω23,(6.7)
a89a199a218a42a7a155a244a234,a183a162a243a89a199a60a111a210a141a116a24a199a99a123a60a89,a51a248a28a162a39a135a232
a73a123a30a127,a24a196a144a139a24a199a56a45,a76a150a24a196a88a52a178a193a210a523a145a56a45,a44a52a183a162a25
a243a139a220a199a56a45,a24a199a52a41a37a158a106a196a123a56a45,a523a145a123;a12a24a199a52a183a1622a145a123a77
a193,a196a49a183a139a24a1992a145a77a193a164a139a24a1993a145a123a56a45,a893a145a56a45a52a199a41a37,a79a36a152
5
a42a24a199a41a37,a28a82a199a198a154a123a155,a183a162a128a199a210a61a107a116a77a193a222a86a234,a89a248a6a210a139
a199a61a107,a25a243a199a139a199a214a35a119a41a14a145a45,a25a243a52a203a177a45a234,a14a145a52a203a177,a139a24
a178a203a177,a13a114a199a123a203a177a196a196a123a56a45a210a52a183a198a117a123a77a193,a183a162a119a198a117a123a77a193a143
a142a56a45,a252a51a24a199a198a154a123a196a139a92a160a35a37a222a210a196a14a145,a149a52a232a196a14a145a45,a199a210
a54a183a162a31a113a145a123,a139a24a30a24a30a123a34,a33a59a92a123a14a145a45a52a199a123a14a145a52a134a34,a0
a143a199a52a134a34a45,a183a65a248a178a199a7a48a196a24a178a77a6,a28a139a80a245a77a6,a154a24a202a77a6a52
a30a134a34,a0a143a139a80a245a77a6,a114a199a77a6a196a24a199a56a45,a89a210a52a183a162a123a14a145a45,a89a52
a134a34a123a60a89,a183a162a25a243a41a123a60a89a52a203a177a45,a89a199a10a39a52a135a232a73a176a62a24a199a99a123
a10a39,a210a52a128,a28a88a52a255a88a24a199a56a45,a13a52a28a243a255a88a220a199a56a45,a63a37a89a220a199a56
a45a131a45a139a183a35a123a7a248,a24a199a52a203a177a196a196a123a56a45,a24a199a52a183a162a123a142a56a45,a15a210
a52a198a117a123a77a193,a89a220a199a56a45a131a45a139a183a196a128a123a89a199a7a248,a89a199a7a248a139a63a135a244a234,
a173a7a244a234,a79a143a139a5a193a123a7a248
dω12 =?Kω1 ∧ω2,(6.8)
a141a30a52a77a193a222a123a42a6,a89a52a79a143Ka52Gaussa77a71,ω1 ∧ ω2a52a193a232a221a222,a196a49
a141a30a52a77a193a222a123a117a159,a38a30a52a24a199a192a220a123a135a73,ω12a52a243a14a145a45E a176a62a123a24
a39a135a73a42,a89a199a24a39a135a73a42a123a105a135a73a127a149a141a30a123a42a6,a89a199a121a210a128a210Gauss
a77a71a144a203Riemann a221a222a139a7,a79a143a7a52a139a234Riemanna221a222a210a139ω12,a0a143a183
a162a141a30a123a42a6a144a203Riemanna221a222a139a7,a89a52Gauss a104a35a105a122a63a123a24a199a136a42,
a203Guassa209a250a122a105a88a122a234a139a89a143a248a6a24a199a7a248,Gauss-Bonneta218a42a210a52a183
a162a7a70a141a30a89a199a42a6a123a232a73,a183a162a25a243a139a24a199a85a20a123a77a193,a199a52a189a53a123,
a7a70a141a30a123a232a73,a70a241a199a123a138a117,a0a143a104a30a183a15a139a24a171a70a216,a79a143a141a30a89
a199a42a6a17a108a52d a24a199a192a220a123a155,a243a24a199a85a20a77a193a222a123a232a73a97a148a520,a47a34
a222,a199a97a148a127a149ω12a215a248a89a199a77a193a123a30a142a123a232a73,a13a152a42a77a193a52a85a20a123,a199
a150a139a30a142,a196a49a97a148a520,a89a23a108a52a70a123,a143a31a143a199a88a127a1490? a183a162a175a108
a139dω12 =?Kω1 ∧ω2,a98a89a199a7a248a88a52a243a24a1992a145a56a45a222,a199a52a243Ea89a1993 a145
a56a45a222,a196a49a183a162a144a21a234a2433a145a56a45a188a126Stokes a189a174,a13a2433a145a56a45a123a155,a89
a199a77a193a2433a145a56a45a176a62a210a139a30a142a234,a28a7a178a89a199a77a193a12a1163a145a56a45a86,a14a143a12
a17? a210a52a154a154a7a201a24a199a252a89a154a41a198a154a123a35a37a222,a166a233a155a128,a89a210a52a196a162a123a37
6
a222a156.a196a49a89a199a77a193a137a7a139a199a37a222a156,a154a154a139a199a35a37a222,a13a89a199a35a37a222a52Ea176
a62a123a24a199a154,a210a178a89a199a77a193a12a116Ea176a62a86,a34a38a139a24a199a77a193,a52a88a52a24a189a139
a199a37a222a156? a89a88a59a92a234,a243a219a92a123a30a127,a104a108a105a59a92,a28a85a5a43a41,a177a34a85a74
a37a222,a210a139a234,a52a88a52a21a234a243a114a199a77a193a201a24a199a37a222a156,a89a52a1a91a176a62a196a162a114
a14a123a175a11,a202a47a95a192a210a180a89a199a175a11,a15a210a52a128,a219a92a23a108a44a49a85a37a222a123,a28
a139a219a92a43a41,a28a178a43a41a73a222a85a5a117,a104a108a210a139a199a37a222a156,a98a52a89a199a52a219a92a123,
a21a88a21a106a223a116a114a199a123a77a193,a88a24a189a44a21,a0a143,a183a222a39a46a178a89a44,a7a89a248a123
a155,a23a140a227a142a89a199a37a222a156a139a73a154(singularity),a7a48a128,a243a5a193a1a199a67a176a62a139
a1a199a37a222a156a123a190a6,( a67a68a63a210a161) a33a38a30a123a190a6,a199a123a73a154a210a243a198a154,a178a44
a89a199a198a154,a53a196a139a48a53a152a37a222,a248a234a198a154a131a105,a210a189a234a24a199a37a222a156,a98a52a198a154
a52a24a199a73a154,a199a52a196a139a121a22a241a117a123a241a21a154,a196a49a199a52a199a73a154,a145a19a199,a196a139a37
a222a209a53a198a154a18,a198a154a164a52a24a199a73a154,a198a154a210a35a196a24a199a188a5a86a123a24a199a154,a93a169
a119sink,a13a38a30a123a119source,a104a108a15a139a54a33a141a30a123a190a6,a44a89a74a190a6a44a49a23a241
a37a222a156a243a73a154a139a88a51a117a159,a152a91a195a85a199a123a88a51a123a117a159,a210a139a24a199a119a41a37a222a156
a123a141a41(index),a28a243a89a24a154,a34a38a139a199a241a193a123a73a154,a0a143a140a248a89a199a73a154a41a199
a66a203a87,a79a143a52a241a193a123a73a154,a196a49a243a66a203a87a222a123a154a123a37a222a52a113a92a104a189a123,a0
a143a25a243,a243a66a203a87a123a154a119a248a73a154a221a245a232a87a17? a152a42a221a24a87,a63a37a52a243a116a123a48
a53a221a24a87,a199a123a141a41a521,a152a42a53a139a123a48a53a210a52?1,a0a143,a243a183a123a222a193a190a6
a165,a195a88sink a164a52source,a141a41a209a521,a118a77a34a123a25a54a141a41a143?1,a73a154a105a133
a236,a79a36a141a41a44a49a82a127a91a138,a34a38a183a178a77a193a12a116a14a145a45a176a62,a12a116a203a177a45a176
a62,a63a37a227a142a139a73a154,a0a143a89a199a222a86a123a77a193a210a139a30a142,a89a199a30a142a210a35a104a149a89
a74a73a154,a196a49a202a226a218a42dω12 =?Kω1 ∧ ω2,a183a162a7a149Gaussa77a71a123a232a73a210a127
a149a73a154a123a141a41a90,a196a49a183a162a121a210a24a199a173a7a117a159,a88a88a28a82a127a91a24a199a37a222a156,
a34a38a199a144a139a139a33a199a73a154,a183a162a89a199a232a73a52a141a41a90,a253a52a178a154a199a73a154a123a141a41a28
a229a117a210a127a149a141a41a90,a89a176a105a7a155,a79a143a89a199a232a73a52a203a37a222a156a160a10a195a7a123,a196
a49a89a121a210a234a24a199a77a193a34a38a139a24a199a139a139a33a199a73a154a123a37a222a156,a243a73a154a123a141a41a90
a37a222a156a123a160a10a195a7,a199a127a149a0a199a232a73,a13a0a199a232a73a176a52a150a139a37a222a156,a196a49a210
a122a116a89a248a24a199a136a42,a183a242a128a24a40,a25a243a139a24a199a85a20a123a77a193,a82a24a199a37a222a156,a139
a139a33a199a73a154,a199a123a141a41a90a52a166a37a222a156a123a160a10a195a7a123,a89a52a79a143a199a127a149a141a30a123
7
a232a73,a13a141a30a232a73a202a253a150a139a37a222a156,a196a49a166a37a222a156a123a160a10a195a7,a143a31a143a89a199
a106a248a127a149Euler a43a117a106a17? a25a243a17a108a199a203a37a222a156a160a10a195a7,a28a210a127a82a24a199a37
a222a156,a7a48a128,a34a38a139a199a77a193,a28a178a77a193a73a191a234,a73a191a196a66a76,a154a199a66a76a52a174
a110a111,a233a149a89a174a110a111,a154a199a30a82a199a123a165a154,a174a110a111a82a199a123a173a101,a28a210a44a49a189
a24a199a37a222a156,a210a54a183a196a152a123,a44a186a154a241a86,a108a128a116a174a110a111a123a173a101a210a159a86,a233a149
a89a248a6a189a123a37a222a156a105a142a52a23a241a117,a166a28a243a30a222a123a89a171a154a123a141a41a127a149?1,a149
a52a199a243a186a154a123a141a41a521,a243a174a110a111a173a101a123a141a41a209a521,a98a52a243a30a222a154a199a154a141a41
a143?1,a196a49a178a89a141a41a28a229a117a123a155,a210a127a149a186a154a123a199a106+a193a123a199a106?a30a123a199a106,
a79a36a210a52Euler a43a117a106,a89a248a6a121a210a234Gauss-Bonnet a218a42.
4 Gauss-Bonneta218a42a123a77a18a249a97a126
Gauss-Bonneta218a42a93a116a139a126a123a30a127a52a77a193a139a30a142,a243a77a193a139a30a142a123a30a127,
Gauss-Bonnet a218a42a52a186a154+a186a154a123a105a110+a30a123a127a143a77a71(geodesic curva-
ture),a242a28a222a243a193a123Gaussa77a71,a5a193a52a24a196a123Gauss-Bonneta218a42
summationdisplay(pi?α) (a154) +summationdisplayintegraldisplay k
g(s)ds (a30) +
summationdisplayintegraldisplay integraldisplay KdA (a193) = 2piχ(M) (6.9)
a233a149a139a30a142a123a77a193,a62a24a92a73a52a30a142a186a154a123a154a77a71,a217a39a52a30a142a123a30a123a34
a77a71,a108a128a114a199a123a89a199a192a220a123a193a77a71,a196a49a28a139a24a199a139a30a142a123a77a193,a28a210
a82a30a142a123a154a77a71+a30a142a123a34a77a71+a193a77a71,a52Eulera43a117a106,a121a210a52a24a248a123.
a93a116Gauss-Bonneta218a42a33a139a126a123a52a139a30a142a123a60a89,a7a48a128,a243a24a199Euclida178
a193,a34a38a139a24a199a174a110a111,a89a199a174a110a111a132a134a34a196a196,a132a149a56a45a52Euclida56
a45,Gaussa77a71=0; a34a38a30a209a52a134a34,a196a49a127a143a77a71a15a520,a79a36a89a199a210a52
a128summationtext(pi?α)a127a1492pi,a89a52a79a143a7a52a174a110a111,Eulera43a117a106a521,a141a30a7a127a1492pi,
a196a49a89a210a128a210a174a110a111a174a110a131a90a243Euclida178a193a222a127a149pi,Gauss-Bonneta218a42a52
a174a110a111a174a110a90a218a42a243a24a196a60a111a123a77a18,a89a199a10a39a173a7a244a234,a199a210a52a114a199a14
a145a45a123a10a39,a183a128,a132a89a199a14a145a45,Maxwella48a199a210a52a89a199a60a89a123a77a18,a28a116
a212a174a222a97a126a123a30a127,a28a123a56a45a524a145,a523a145a56a45+1a145a30a45,a524a145a123a98a85a251a22
a111,a0a143a7a44Maxwella48a199a123a155,a28a7a126a24a199a203a177a45,a34a19a222a52a24a199a133a123a134
8
a34a45,a199a139a199a77a71,a183a162a123a77a71a52Gaussa77a71a198a49a193a232a195a163,a13a89a199a77a71a52
a1992a39a135a73a42,a178a44a43a89a199a77a71a52a85a20a123a30a71a85a241a117a210a52Maxwe lla48a199,a196
a49,Maxwella48a199a123a1a91a242a181a52a58a158a59a92a123,a210a52a79a143a45a142a524a145a123a56a45,a196
a49a52a442a145a56a45a106a223a1164a145,a0a143a89a199a77a71a79a143a52a24a1992a39a135a73a42,a164a52a39a233
a193a123,a79a36a2434a145a56a45a176a52a24a1994×4a48a106:
(Fαβ =
0 E1 E2 E3
E1 0?B0 B2
E2 B0 0?B1
E3?B2 B1 0
,(6.10)
a89a199a48a106a176a62E1,E2,E3 a52Electric Potential,B0,B1,B2a52Magnetic Po ten-
tial,a15a210a52a158a51a203a30a51,a89a74a209a52a48a106a176a62a123a60a106,a44a43a132a89a199a48a106a196a44a43
a1232a39a135a73a42a52a85a20a123,a253da89a199a42a6a123a135a73a1430,a210a52Maxwell a48a199
d(Fαβdxα ∧dxβ) = 0,(6.11)
Gauss-Bonneta218a42a52a24a199a187a36a222a105a245a220a35a123,a93a116a183a85a123a218a42a17a88a132
a149Gauss,a15a88a132a149Bonnet,Gaussa144a41a234a174a110a111a123a60a89,a132a127a143a174a110a111a41a116
a174a110a111,Bonneta15a150a139a41a95a192a123a97a126,Bonneta178a174a110a111a77a18a116a127a63a77a34,a198
a178a127a63a77a34a123a127a143a77a71a232a73a44a143Gauss-B onneta218a42a123a232a73,a104a35Bonneta52
a27a41a33a7a155a123a1a91a166a27,a198a243a135a73a1a91a41a234a58a158a228a253a123a224a26,a183a88a11a28a162
a234a137a245a232,a183a230a130a28a162a234a137a89a24a92a73a123a106a166a58a158a7a155,Maxwella48a199a52a199a123
a1a47a60a89,a89a199a58a158a139a126a255,a183a139a24a159a169a57a243a152a35a123a21a41a166a22a222a21a44,a11a248
a119a21Gauss- Bonneta218a42a166Maxwella48a199a22,a183a89a123a142a245a7a154a243a89a159a169a57a176a62
a139.
9
2001a3510a21911a134
1 a135a232a73a123a229a205,a58a238a166a116a89a26a251
a89a116a135a232a73,a33a7a155a123a220a199a124a52a58a238(Issac Newton,1642-1727)a203a116a89a26
a251(Gottpied Leibniz,1646-1716),a135a232a73a210a52a198a162a21a25a123,a7a149a58a238,a139a108
a85a123a52a198a41a89a199a211a42a52a243a166a9a123a30a127,a15a142a7a28a162a123a181a106a164a7a66,a0a199a30a127,
a15a210a5217a45a22a0a199a30a127,a78a179a166a60a105a184a51,a78a179a139a234a105a245a124,a198a243a93a41a74a24
a76a166,a79a143a166a60a123a7a248,a166a68a56a34a234,a198a210a195a27a243a27a176a41a7a149a135a232a73a123a89a74
a211a42,a116a89a26a251a52a24a199a200a48a193a209a58a158a128a132a123a124,a106a166a52a198a123a108a85a123a24a92a73,
a198a123a108a85a116a13a115a1a27a70a200a48a193a209a139,a198a162a220a124a131a45a139a154a112a88,a52a79a143a112a88a120
a52a135a232a73a123a21a25a86,a89a199a112a88a52a88a115a123,a15a150a139a31a143a63a66,a34a159a222a52a116a89a26
a251a62a24a199a21a44a7a149a135a232a73a88a169a123a124,a198a123a88a169a2431684a35a21a44,a58a238a41a89a199a211
a42a0a149a116a89a26a251,a13a116a89a26a251a21a44a88a169a0a149a58a238,a58a238a139a234a89a199a211a42a128a150a139
a21a44a31a143a127a91a123a192a220,a13a116a89a26a251a88a98a21a44a234a89a74a192a220,a51a30a164a90a126a234a24a74
a110a82,a15a142a183a162a25a243a164a243a126,a0a143a128a117a220a199a124a139a24a199a112a88,a76a150a209a52a203a106a166
a150a139a7a248a123a124a243a0a176a5a196a123a60a89,a89a88a52a24a199a31a143a139a63a135a123a47a60.
2 a135a232a73a228a253a189a174
a135a232a73a52a106a166a176a62a105a173a7a123a48a193,a150a149a31a143a52a135a232a73a17a218a183a46a135a73a123a21a25
a203a136a11a14a21a25a43a41a58a158a139a7a248,a79a143a136a11a14a21a25a43a41a131a128,a106a166a204a7a123a248
a123a210a52a207a196a60a106,a207a196a220a28a106a123a7a248,a139a171a171a123a7a248,a183a162a127a119,a60a106a139a171
a171,a139a34a117a123,a58a34a117a123,a174a110a60a106a127a171a171a60a106,a0a143a7a14a143a248a143a207a196a60a106a123
a117a159? a183a162a209a127a119,a60a106a44a49a126a77a34a117a44a43,a152y = f(x) a89a30a77a34,a243a89a30
a77a34a123a154a154,a152a42a199a52a44a49a135a73a123a155,a0a143a199a243a154a154a139a199a35a34,a135a73a210a52a178
a89a199a77a34a126a199a123a35a34a117a207a196a199a123a117a159,a196a49a15a127a149a128a199a52a178a60a106a34a117a154,a34
1
a117a154a131a128,a44a49a28a1a62a1a198a1a248,a44a49a15a174,a79a36a44a49a122a116a106a241a117,a106a166a7
a52a21a234a122a116a106a241a117,a15a52a105a7a155a123,a196a49a135a73a76a150a52a128a126a77a34a123a35a34a117a207
a196a77a34a123a117a159,a232a73a117a122a0a234,a79a143a232a73a34a19a222a76a151a89a229a117,a199a52a7a15a174a193
a232,a0a143a34a38a178a193a222a139a24a199a75a173,a132a77a34a117a41a143a30a142,a199a123a193a232a139a245a76,a203a177
a123a193a232a139a245a76,a89a176a123a175a11a52a232a73a123a13a41,a15a52a232a73a173a7a123a248a123,a79a36,a34
a19a222,a232a73a123a21a48a243a135a73a131a3,a232a73a104a30a15a150a139a24a189a123a189a66,a232a73a210a52a139a199
a244a33a123a10a39,a77a34a196a140a194a123a75a173a24a196a46a27a6a126a134a34a117a5a163,a38a122a5a163a123a77
a34a74a149a28a123a30a142a123a30a127,a210a139a199a244a33,a210a52a89a199a75a173a123a193a232,a196a49,a15a13a211
a131,a232a73a123a21a48a243a135a73a131a3,a165a45a89a220a199a175a11a80a54a150a139a7a248,a98a52a217a34a89a7
a248a58a158a123a183a35,a232a73a139a88a245a52a135a73a123a39a228a174,a7a48a128,a34a38a28a70a89a30a134a34a203
a220a30a18a34a196a196a75a173a123a193a232,a89a220a30a18a34,a24a199a52s = a,a24a199a52s = x,a28a7a86
a174a89a199a75a173a123a193a232,a52a199a189a232a73integraltextxa f(x)dx,(a214a42f(x)a189a232a73a44a → x),a89
a52a104a35a116a89a26a251a123a110a82,a89a199a232a73a123a110a82a16a196a89a248,a79a143a232a73a15a52a83a44a24
a199a90,integraltexta83a44a90(sum),a34a247a193a232a24a30a132s = a a123a134a34a42a30a142,a12a24a30a52a127a63
a123x,a28a178xa89a30a134a34a35a196a123a155,a210a122a116a24a199xa123a60a106,a89a199a60a106,a183a119a199A(x),
a210a52a183a67a222a123a193a232,a52a199a232a73,a196a49a199a52a24a199a106a248,a166xa139a7,a196a49a52xa123a60
a106,a89a199a60a106a203a77a34a48a199y = f(x)a89a199a60a106a139a183a35a7a248,a143a31a143a139a183a35a123a7
a248a17a218a105a59a92a143a23a23,a34a152a70A(x) = integraltextxa f(x)dxa123a135a73,a70a199a123a135a73a108,a210a52
a128,a70s = x,x + δx a196a140a196a123a89a199a66a30a75a173a123a193a232,a25a243a152a42a28a252δxa248a123a155,
a183a46a105a142a52a23a241a117a234,a89a199a244a33a210a52f(x),a196a49a105a142a52a23a241a117A(x)a89a199a60a106
a123a135a73a210a52f(x),a79a36
dA(x)
dx = f(x),(1.1)
a89a210a52a135a73a51a232a73a123a228a253a123a7a248,a89a199a7a248a128A(x)a52a24a199a232a73,a70a199a123a135a73
a123a30a127,a210a122f(x),a89a199a24a196a143,a119a41a135a232a73a123a228a253a189a174,a183a44a3a243a8a13a39a135
a232a73a123a30a127,a41a170a88a195a143a31a143a89a52a24a199a135a232a73a123a228a253a189a174,a79a143a24a196a143a178a89
a199a7a248a42a85a196 integraldisplay
f(x)dx|ba =
integraldisplay b
a
f(x)dx (1.2)
a111a231,a38a30a232a73a52a199a88a189a232a73(indefinite integral),a88a189a232a73a52a199a60a106,a38a42
2
a52a60a106a243ba123a138a62a86a60a106a243aa123a138,a127a149a89a199a189a232a73(definite integral),a196a49
a44a89a199a7a248a127a119a7a70a232a73a123a155,a144a137a7a70a24a199a60a106,a199a123a135a73a52a46a127a123,a210
a52f(x),a253a135a73a52a46a127a123,a196a49a89a248a135a73a203a232a73a203a229a117a234,a144a35a123,a232a73a127
a149a135a73a123a39a228a174,a139a234f(x),a7a73a24a199a60a106,a199a123a135a73a127a149f(x),a52a199a39a228a174.
a79a36a135a1a232a73a139a183a35a123a7a248.
3 a245a195a135a232a73
a222a193a89a123a52a24a199a35a106a123a135a232a73,a5a193a89a176a145a123,a7a245a35a106a123,a245a35a106a123a155,
a139a99a123a25a54,a52a31a143a248a123a17a218a183a46a233a149a245a35a106a123,a183a162a11a88a23a47a123,a11a23a220
a199a35a106a123a60a111,xa203y,a0a143a183a162a127a119a89a199a30a127a135a73a123a10a39a123a77a18a52a160a135a73,
a127a149xa203ya73a13a70a135a73,a232a73a123a10a39a77a18a52a173a232a73,a19a173a232a73(double integral)
a52a2432a145a123a60a111,a243a176a145a123a60a111a52a245a173a123,a11a232a145,2a145a123a60a111a210a139a234a75a173,
a183a162a119a199?,a0a143a199a123a30a142a119a199γ,a196a49a232a73a123a24a199a7a108a77a18a52a24a1992a173a232a73,
a202a47a232a73a178xa73a196a66a227,a108a128a82a66a227a242a198a222a89a199a60a106,a70a24a199a90,a2432a173a232a73
a123a30a127,a48a27a15a52a178a75a173a73a196a66a76,a108a128a82a154a24a66a76a123a193a232,a243a217a222a60a106a138
a198a222a199a123a193a232,a108a128a70a199a123a90,a105a88a122a234a123,a34a38a60a106a80a123a155,a195a88a28a152a91a87
a28a123a75a173,a244a33a52a24a248a123,a196a49a89a244a33a210a522a173a232a73
I =
integraldisplay integraldisplay
f(x,y)dxdy,(1.3)
a2432a145a123a30a127,a4a150a176a145a123a30a127,a24a199a173a7a123a25a54a52,a183a162a25a243a1392a199a35a106x,y,
a166a35a106a14a143a248a218a196a49a183a25a243a166a35a106,a166a35a106a104a108a52a243a135a232a73a176a52a105a173a7a123
a24a199a205a27,a79a143a105a245a123a175a11a52a23a28a123a35a106a52a100a160a10a122a55a104,a139a30a166a35a106,a175
a11a210a193a47a59a92a154a234,a210a44a49a137a251a234,a25a243a183a166a35a106a213
braceleftBiggx = x(xprime,yprime)
y = y(xprime,yprime) (1.4)
a217a165,(xprime,yprime)a52a12a105a24a28a43a41,a183a162a21a25a24a199a47a34,a243a176a145a123a30a127,a135a73a123a198
a27,a183a162a85a196dx ∧ dy,a89a52a24a199a198a27,a14a143a198a17a218dx ∧ dya243a135a232a73a222a52a33a135
a201a123a10a154,a31a143a119a135a73a218a31a143a52dxa218a89a199a52a104a118a234a106a166a27a1a186a35a123a47,a14a143
3
a248a189a135a73a123a189a66a203a196a190a31a143a52dx,a89a199a105a102a38,a44a49a41a116a105a119a63,a88a44a178a199
a89a56a249a137a7a139a24a189a123a30a45,a196a49a183a106a106a141a141a128a139a24a199dx,a243dx,dya89a171a135a73
a131a45a7a79a193a198a27∧,a31a143a119dx ∧ dya218a89a199a175a11a205a133a236a234,a28a152a42dx,dya253a252
a52a31a143a209a88a56a249,a198a234a49a128a52a31a143a192a220a205a52a24a199a105a135a201a104a10a123a175a11,a243a89a48
a193a139a24a199a76a123a159a90,a210a52a90a159a105a83a106a90a105a135a73,a34a189dx ∧ dya89a199a198a27a52a39a233
a193,
dx∧dy =?dy ∧dx,(1.5)
a89a199a175a11a210a56a249a59a92a234,a79a143a198a27a152a42a52a39a233a193a123a155,a104a108dx ∧ dx = 0,a47
a34a222,a79a143dx ∧ dx =?dx ∧ dx,a196a49dx ∧ dx = 0,a243a39a233a193a123a198a27a131a5,
a178dx ∧ dya23a196a35a106,a79a143a198a27a52a39a233a193a123,dx2 = 0,a196a49a210a150a139a176a39a123a192a220
a234,a89a248a122a116a123a83a106a119a41a105a83a106,a89a199a83a106a105a201a123,a139a24a199a193a47a123a136a88a213a166
a35a106a218a42a143
dx∧dy =?(x,y)?(xprime,yprime)dxprime ∧dyprime,(1.6)
a34a38a183a162a123a135a73a126a123a52a160a135a73,a196a49
dx =?x?xprime dxprime +?x?yprime dyprime,dy =?y?xprime dxprime +?y?yprime dyprime,(1.7)
a25a243a126a105a198a27a24a198,dxprime ∧ dxprime = dyprime ∧ dyprime = 0,a13dxprime ∧ dyprimea79a143a198a27a52a39a233a193
a123,a196a49a52a166a80a198a49x = x(xprime,yprime),y = y(xprime,yprime) a123a196a44a7?(x,y)?(xprime,yprime),a89a199a110a82a52
a196a44a7,a52a143a199a160a135a73a196a196a123a113a239a42,a196a49
dx∧dy =?(x,y)?(xprime,yprime)dxprime ∧dyprime,(1.8)
a89a199a166a28a52a183a162a173a232a73a166a35a106a123a24a199a7a248.a183a162a127a119a173a232a73a7a52a166a35a106a123
a155,a199a97a148a198a222a196a44a7,a196a49a89a199a136a88a210a52,a233a173a232a73a123Integral,a253a232a73a5
a123a42a6,a178a232a73a82a191a171,Integrala52a24a199a135a73a245a49a42,a198a27a52a39a233a193a123,a196a49
a34a38a245a173a232a73a1393a145,4a145a116n a145a123a56a45,a245a173a232a73a123Integrala44a23a196a52a105a83
a106a123a245a49a42,a0a143a166a35a106a210a7a108a233a234,a89a176a62a139a24a154a135a201a123a143a48,a79a143a47
a158,a28a7a121a210a166a35a106a123a218a42a123a30a127,a34a189a196a44a7a52a116a123,a88a108a123a155,a198a222a196
a44a7a123a253a233a138,a38a199a52a116a123,a89a199a52a176a145a1a91a135a201a123a192a220,a210a52a56a45a139a199
4
a53(Orietation),a28a221a123a30a127,a1392a199a35a39a221a123a48a53,a221a123a30a127,a34a38a149a234a48a53
a123a155,a196a44a7a52a139a138,a79a36a183a162a24a199a136a88a52a245a173a232a73a123Integrala97a148a52a24a199a105
a83a106a245a49a42,a52dx,dya123a245a49a42,a198a27a52a39a233a193,a89a248a166a35a106a113a92a44a49a233a123,
a104a108a183a144a41a2342a145a123a190a6,a176a145a52a105a210a23a123,a51a248a123.a105a198a27a52a201a122a105a254,a52
a88a204a139a176a39a123,a196a49a7a118a59a92,a178a48a24a5,a210a520.
4 a105a135a73
a222a193a89a234a89a143a248a24a171a7a248,a4a150a89a7a248a164a205a7a80,a183a162a89a176a127a135a232a73a123a30a127,
a24a199a173a7a123a189a174a52a194a245a189a174(Green’s Theorem),a210a52a128,a34a38a28a139a199a75a173,
a243a30a142a222a123a135a73a52a44a49a35a143a75a173a222a123a135a73,a52a24a199a24a173a232a73a90a19a173a232a73a123
a7a248,a89a52a199a58a158a173a7a123a7a248,a7a48a215a12a115a71a139a24a253a66a86,a89a116a89a199a7a248,a198
a128a143a89a52a114a199a135a232a73a123a228a253a189a174,a183a52a51a63a123,a89a248a123a7a248a25a243a47a158a85a194a245
a189a174a123a30a127,a128a128a52a85a196a139a232a73,
integraldisplay
γ
Adx + bdy = (?B?xA?y )dxdy,(1.9)
a152a42a139a24a199a175a11,a139a30a127a28a44a49a144a11Integral,a88a7a11a217a199,a0a143Integrala210a52
a178a24a199a24a39a135a73a42a35a143a220a39a135a73a42,a89a14a143a35a17a218a218a42a189a174a52a89a248a6,a183a210
a90a156a24a199a105a135a73,a183a162a166a98a89dx ∧ dy a52a24a199a245a49a42,a52a24a199a105a83a106a123a24a199
a42a6,a210a54a183a162a202a47a245a49a42a24a248,a88a98a152a36,a233a149a89a248a123a42a6,a183a162a164a44a49a189
a66a199a24a199a135a73,
d(Adx + Bdy) = dA∧dx + dB ∧dy = Aydy ∧dx + Bxdx∧dy,(1.10)
a119a105a135a73(Exterior differential calculus),a105a135a73a105a59a92,a34a247a139Adx + Bdy,
a199a123a135a73a210a52a135a73a199a123a248a106,a15a210a52a135a73a60a106,Aa166Ba52x,ya123a60a106,a196a49
a210a135a73A,B,Aa123a135a73a210a52Axdx + Aydy,Ba123a135a73a210a52Bxdx + Bydy,a44
a52Axdx∧dx = 0 a210a122a116Aydy ∧dx,a145a19a49a210a122Bxdx∧dy,a98a52a79a143a198a27a52
a39a233a193a123,a196a49a210a122(Bx?Ay),a89a52a194a245a189a174a176a622a173a232a73a123a248a106,a196a49a194a245
a189a174a178a92a39a232a73a35a196a220a39a232a73,a199a123Integrala34a19a222a52a199a105a135a73,a44a49a23a241a105
5
a135a73a52a105a201a123a192a220,a79a36a28a44a49a178a232a73a82a191a171,a210a128a183a162a252dx,dya5a24a199a105
a83a106,a233a89a199a105a83a106a139a199a105a135a73,a105a135a73a105a59a92,a210a52a34a38a135a73a200a49a123a30a127,
a217a34a52a233a154a49a248a106a135a73,a136a42a183a122a116a24a199a245a49a42,a89a199a245a49a42a123a39a106a176a24
a199,a42a143a60a106a210a35a143a24a39a135a73a42a234,a196a49a39a106a176a24a199,a79a36a210a42a143a198a117a52ka39
a123a155,a122a116a24a199k + 1a39a123a135a73a42,a89a199a52a194a245a189a174a165a152a91a178a77a34a135a73a123a135a73
a42a35a143a75a173a135a73a42,a24a173a135a73a35a143a19a173a135a73a123a218a42,a89a199a210a105a80a234,a79a143a89
a176a193a139a24a199a105a83a106,a196a49a178a89a199a135a73a42a198a229a117,a126a24a199a105a198a27,a135a73a123a198a27
a52a39a233a193,a108a128a17,a25a243a183a139a24a199a135a73,a199a178ka39a123a105a135a73a42a35a143k + 1a39a123
a105a135a73a42,a89a248a6a210a178a89a199a105a135a73a42a165a45a201a234a24a199a99a123a136a232,a44a49a135a73,a89
a199a135a73a203a202a47a123a135a73a88a24a248,a199a52a178ka39a35a143k + 1a39,a135a73a24a196a143a15a52a28a24
a39,a89a199a105a135a73a52a33a0a30a127Frobenius,Daubouxa90a183a123a144a19Elie Cartana90a159
a117a123,a198a162a33a240a90a159a89a199a10a39a52a233a149a24a39a135a73a42,a52Frobenius,Daubouxa90
a156a123a24a39a135a73a42,a13Elie C artana52a27a41a123a115a71,a52a183a123a144a19,a198a57a89a52a19
a27a45a22,a15a210a52a222a199a45a22a33a149a76a123a1a91a166a27,a27a41a174a169a76a166a123a115a71,a183a46a89
a171a115a71a105a52a220a41,a198a88a41a47a123a217a196,a219a41a106a166,a30a158a213a49a52a113a92a99a123,a139a24
a35,a198a201a234a24a160a49,a52a21a137a219a197a166a22(Analytical Mechanics),a198a178a105a135a73a123
a10a39a44Frobenius,Daubouxa44a24a39a42a123a189a66a77a18a116a176a39a42,a196a49a114a199a123a105a135
a73a52Elie Cartana90a159a117a123,a89a52a139a126a123a192a220,a89a199a105a135a73a139a219a5a123a25a54,a52
a126a220a39a131a128a127a1490.
d2 = 0,(1.11)
a253a89a199a105a135a73a126a220a39a127a1490,a183a162a7a121a210(1.11),a210a52a233a195a88a24a199ka39a135a73a42,
a135a73a24a39a210a35a143k + 1a39,a220a39a210a35a143k + 2a39a135a73a42,a199a24a189a520,a7a121a210a89
a24a154,a183a121a210a233a149a60a106a233a234,a210a113a234,a196a49a183a7a121a210a233a149a127a63a123a60a106f,a178
a89a199d,a105a135a73a126a220a39,a210a127a1490,a253d2f = 0 a210a113a234,a0a143a143a31a143a17a218a79a143
a23a108a183a7a121a210d2 = 0,a144a7a121a210d2a42a126a243a144a139a24a49a222a233a210a113a234,a89a52a79a143a199
a52a34a117a123,a196a49a152a42a34a117a24a49a139a89a199a117a159,a0a143a114a199a123a90a210a127a1490,a0a143a24
a49a123a155,a209a52a24a199a60a106a198a222a24a28dx,a183a25a243a160dxi,a210a52a34a189a243a176a145,a243na145,
xia210a52x1a116xn,a243a176a145a30,a152a42a139a24a199a60a106f,fa52x1,···,xna123a24a199a60a106,a233a149
a89a199a60a106,a126a105a135a73a220a39,a24a189a127a1490,a47a34a222,a79a143a105a135a73a24a39a210a122a116aia52fi
6
a233xia123a24a199a160a135a73,a0a143a242a126a24a39a17,a199a123a248a106a210a52a44xi a116xja135a73ai,ai
a52fa123a233xi a123a135a73,a196a49a89a52fa233a44xia116xja123a19a126a135a73:
d(aidxi) =?ai?x
j
dxj ∧dxi,(1.12)
a89a199a60a106a233a149i,ja52a233a193a123,a47a34a222a183a162a127a119a24a199a60a106a135a73a220a39a123a155a203a39
a147a150a139a7a248,a52a233a193a123,a152a42a24a199a233a193a123a60a106a52dx ∧ dya123a248a106,a13dx ∧ dya52
a39a233a193a123,a0a143a199a210a127a1490 a234,d a52a24a199a105a135a73,a52a233a105a83a106a123a245a49a42a123a24
a199a228a174,a89a199a228a174a228a126a220a39a210a127a1490 a234,a89a52a24a199a234a88a122a123a7a248,a79a143a1a91
a222a89,a34a38a28a139a24a199a75a173,a28a82a89a199a75a173a123a30a142,a242a82a89a199a30a142a123a30a142,a210
a150a139a30a142a234,a34a38a28a82a123a30a142a52a114a199a69,a0a143a69a150a139a30a142,a196a49a1a91a222a89a139
a24a199a228a174a70a30a142,a70a30a142a123a155,a126a220a39,a210a127a1490,a139a24a199a75a173a123a70a24a39a30a142
a52a24a199a105a80a123a75a173,a253a88a242a139a30a142a234,a89a199a1a91a123a117a159a203a105a135a73a123a117a159a52a233
a83a123,a70a220a39a30a142a24a189a127a1490,a89a52a199a1a91a123a117a159; a70a105a135a73a220a39a127a1490,a52
a199a73a219a123a117a159,a89a220a199a192a220a88a52a220a199a144a88a35a7a123a192a220,a52a113a92a233a83a123,a52a24
a195a47,a24a199a30a142a47a158a126a110a82?a44a43,a30a142a220a39a127a1490,a253?2 = 0,a199a203a105a135a73
a52a233a83a123,a89a52a24a199a234a88a122a123a1a91a7a248,a234a88a122a123a106a166a222a123a7a248,a201a122a88a122
a234,a79a143a70a30a142a52a24a199a1a91a123a175a11,a205a52a24a199a114a14a123a175a11,a24a189a7a252a114a199a75
a173a198a222a30a142,a98a52a70a105a135a73a52a199a73a219a123a175a11,a52a199a219a92a123a175a11,a7a105a135a73a144
a7a127a119a89a199a135a73a42a243a24a154a142a163a123a117a159a210a139a234,a89a24a199a219a92a123a228a174a203a24a199a114
a14a123a228a174a139a89a248a233a83a123a7a248a52a105a10a122a123a47a60,a52a24a199a173a7a123a1a91a25a54,a52a173
a7a123a106a166a25a54,a143a31a143a233a83a17a218a217a34a89a210a52a194a245a189a174a123a77a18,a210a52Stokesa189
a174,Stokesa189a174a89,a34a38a139a24a199a75a173,a178a199a85a20a222,?ka52a89a248a24a199ka145a123a75
a173,a196a49a199a123a30a142a210a52a30a142k,a0a143a34a38a139a24a199a135a73a42a119a41α,a199a123a39a106
a52k? 1(degα = k? 1),a149a52a183a162a210a139a89a143a24a199a7a248,α a243a30a142a123a232a73a127
a149dα a243?ka123a232a73,integraldisplay
k
α =
integraldisplay
k
dα,(1.13)
a89a52a173a7a244a234a123a189a174,a47a158a126Stokes a214a66,Stokes a52a93a41a123a97a126a106a166a27,a28
a162a76a150a243a89a199a49a165a46a178a38a116Stokes a189a174,Stokes a189a174a210a178a220a199a202a47a123a228a174,
7
a24a199a52a127a149a75a173a123a30a142a123a228a174,a24a199a52a127a149a105a135a73a123a232a73,a89a220a199a139a59a92a123
a7a248,a34a38a183a162a178a105a135a73a123a232a73a85a196a89a199a7a248,
(,α) = (?,dα),(1.14)
a89a199a105a135a73a196a24a199a37a222a56a45(Vector Space),a44a49a28a62,a89a199a75a173a15a52a12a105a24
a199a37a222a56a45,a15a44a49a28a62,a34a38a89a220a199a37a222a56a45a178a44a232a73,a79a36a210a139a24a199a196
a162a123“a233”(pair),a89a199a37a222a56a45a123a24a154a90a0a199a37a222a56a45a24a154a203a243a24a229a52a122a116
a24a199a116a106,a122a116a24a199a106,a0a143St okesa189a174a210a52a128a89a199paringa38a122a233?a123a42a126
a123a174a6? a166a105a135a73d a52a202a177a123(adjoint),a52a233a83a123“a233”,a89a210a52Stokesa189a174a123
a63a66,a176a145a30,a249a127a63a145a30a209a52a233a123.a215a12a115a71a243a198a123a66a86a176a128,a89a199a52a135a232
a73a123a228a253a189a174,a44a199a210a201a241a183a162a202a47a135a232a73a123a228a253a189a174,a79a143a34a38k = 1,
a0a143a183a162a123a75a173a52a24a199a34a227,a44a a116ba123a34a227,a89a199a34a227a210a52?,a199a123a30a142a17,
a52ba154a62aa154,α a243a89a176a52a24a199a60a106,a222a39a89a123dαa52a199a232a73,a243a24a145a123a60a111a210
a52a126a116a134a34a222,a79a36a243a24a145a123a60a111?a52a199a34a227,a199a123a30a142a52b? a,αa52a24a199a60
a106f,a196a49dαa52df,a149a52
(b?a,f) = (?,df) =? f(b)?f(a) =
integraldisplay b
a
df,(1.15)
a89a210a52a128a60a106a243ba154a123a138a62a86a60a106a243aa154a123a138a127a149dfa243a89a199a34a227a222a123a232a73,
a89a199a210a52a196a162a135a232a73a123a228a253a189a174,a15a210a52a128a141a30a52a44aa116ba232a73df,a38a30a210
a52f(b)? f(a),a89a210a52a183a162a123a228a253a189a174,a196a49Stokesa189a174a52a135a232a73a123a228a253a189
a174a243a176a145a123a77a18,a79a36a243a245a195a123a135a232a73a176a62a15a52a199a159a90,a58a158a139a126,a79a143a105
a135a73a221a57a105a245a97a238,a139a24a199a218a42a105a142a52a121a210a123,a210a52a28a178a220a199a105a135a73a123a42
a6αa203βa35a198,a13a70a89a199a123a105a135a73,
d(α ∧β) = dα ∧β + (?1)degαα ∧dβ,(1.16)
a89a199a218a42a105a142a52a121a210,a79a143a59a92a143a144a7a34a189αa90βa209a92a49a210a113a234,a89a52a132a149a233
a149αa90βa209a52a34a117a123,a34a189a199a162a209a52a92a49a123,a210a44a49a85a196dx1,···,dxk,···,dxn,
a3a62a198a199a60a106a24a174a210a44a49a122a116a234,a196a49a199a162a89a199a198a27a131a45a90a105a135a73a139a89
a248a24a171a59a92a123a7a248,a89a199a7a248a88a98a152a36,a164a44a49a205a207a123,a79a143a34a38a139a24
8
a199a228a174,a199a123a178a48a127a1490,a89a52a105a88a122a234a123,a89a199a210a44a49a5a24a199a248a27,a139a199
a219(quotient),a89a248a122a116a24a199a248a27,a25a243a119a41a51a174(homology),a25a243a142a245a106a166
a123a21a48a209a52a139a199a228a174,a28a220a39a127a1490,a28a210a21a5a24a199quotien t,a14a143a248a17,a31
a143a119quotienta17? a210a52a28a178a196a139a123a119a22dα = 0a123α,a250a196a139dβa117a248,a253
{α|dα = 0}/dβ,(1.17)
a7a52α = dβa123a155,a79a143d2 = 0,a196a49dα = 0,a79a36a28a82a196a139a123a196a162a123a20a111
a42(close form),a250a44a49a85a196da31a143a123a192a220a117a248,a210a122a116a243a106a166a176a62a126a24a199a142
a124a123a214a9a119homology,a15a210a52a82a196a139a123ka39a123a135a73a42,a199a162a52a85a20a123(a250da42
a126a1430),a250a196a139a123dβa123a248,a5a24a199a219a136a232,a89a199a219a136a232a210a119a41homology,a28a44
a49a126a116a89a199d,a15a44a49a126a116a89a199a30a142,a126a116a30a142a123,a187a36a222,a52a243a95a192a176a62,a11
a139a126a30a142a123,a79a143a126a123a52?a123homology a119a222a51a174(cohomology),a89a52a132a149a187
a36a123a7a248,a214a9a126a171a234,a196a49a119cohomology,a89a199a105a184a51,a34a38a28a139a24a199a22
a111,a199a52a155a151a123,a199a123cohomology forma52a139a33a145a123,a89a199a139a33a145a123a145a106a119a89
a199a56a45a123Betti a106(Bett i Number),a89a52a95a192a123a19a142,a92a166a135a232a73,a44a49a88a23
a86a11,a88a44a89a199a11a173a114a199a123a139a173a7a123a21a48,a52a163a117a106a166a123a21a48a228a253a19a142,a104
a108a105a7a155a234,a28a139a24a199a105a76a123a56a45,a196a139a135a73a42a28a196a123a56a45a76a122a88a122a234,a199
a139a136a232,a28a44a49a28a62,a15a44a49a70a105a135a73,a76a122a88a122a234,a108a128a17,a199a139a74a1a91a123
a117a159,a82quotient,a89a199quotienta52a139a33a123,a89a199a139a33a139a199a80a255,a122a116a106a248a139a33,
a52a128a139a33a145a123a145a106a52a245a232,a122a116a24a28a106,a89a28a106a248a210a52a89a199a56a45a123a173a7a117
a159,a79a143a122a127Betti a106a52a24a199a114a106,a139a24a107a114a106a105a7a155,a7a48a128,a69a193,a69a193a139
a89a171Bet ti a106,a162a193a15a139Betti a106,a199a162a52a88a24a248,a5a193a180a95a192a123a124a46a27a7a121
a210a89a171Betti a106a52a95a192a88a35a222,a79a36a95a192a243a106a166a123a228a126a165a210a7a155a234.
9
a145a19a89 a141a106a166a233a106a60a106
2001a3510a21919a134
1 a253a49a123a15a153a90a248a123
a164a139a1a73a168,a183a46a190a89a199a229a204a89a24a89a183a123a15a153a90a248a123,a183a89a199a49a123a49a30
a528a199a66a30,a98a135a232a73a76a122a88a122a234,a135a232a73a123a41a140a105a18,a88a7a1288a199a66a30,a210
a5280a199a66a30a15a89a88a113a123,a196a49a183a104a108a144a21a89a199a76a150,a131a217a52a144a235a114a199a123a139
a24a74a63a66a123a175a11,a150a149a45a251a123a60a111a183a150a27a86a245a89,a88a45a251a123a189a66a221a86a121a210,
a183a46a28a162a46a178a166a44a135a232a73,a196a49a183a209a88a24a189a7a201a28a162a107a31a86,a28a162a195a86a23
a24a23a7a3a49a3a126a123a86,a76a150a243a86a176a73a122a116,a15a139a183a123a89a123a41a140a90a19a142a52a86
a165a150a139a123,a183a250a122a97a148a10a24a10a135a232a73a114a199a123a107a47a221a86a52a243a0a74a48a193a53a3a21
a48,a44a49a128,a135a232a73a53a3a21a48a76a150a139a220a199a33a173a7a123a48a193,a24a199a52a243a1a91a123a97
a126,a135a232a73a243a135a73a1a91a123a97a126,a33a0a52Gauss,Gaussa15a142a88a52a33a0a123,a97a148
a164a139a47a123a124,a152Euler,Mongea127a124,a88a44,a183a46Gaussa5219a45a22a92a45a142a33a149a76
a123a106a166a27,a106a166a243a0a30a127,a92a45a142a15a210a52a220a78a234,a79a143a89a199a198a79,a121a41a123a106
a166a24319a45a22a52a92a45a142a33a80a123,a0a30,a88a98a139Gauss,a164a139Gaussa123a107a47a249a217a166
a9,Gaussa33a7a155a123a166a9a210a52Riemann,a79a143a139Gaussa90Riemann,a121a41a123a106
a166a210a11a11,a11a11a123a63a135a210a52a76a27a203a248a198a123a48a53a86a21a48,a243a1a91a222a123a97a126a123a21
a48a52a105a245a123,a104a35Einsteina17a128a44a212a174a25a54a210a52a1a91a25a54,a49a36a21a48a198a123a18
a66a35a233a88,a18a66a35a233a88a104a108a7a126a43a41,Einsteina234a137a33a240a123a43a41a44a43a1a91a175
a11,a230a130a43a41(x,y) a139a1a91a123a63a66,a104a24a199a212a174a166a27a250a122a97a148a139a1a91a123a221a212
a174a123a63a66a30,a198a41a229a117a98a7a118a92a174,a88a44,Einsteina122a122a234a137a89a199a41a88a116,a79
a143a56a45a17,a117a122a7a118a133a236,a199a227a142a127a63a43a41,a227a142a43a41a123a127a63a160a10,a79a36a15a227
a142a43a41a35a166,a89a210a52a183a162a25a243a196a119a123a22a111,a22a111a123a150a39a52a56a45a150a39a123a77a18.
a253a117a126a123a52Euclida56a45a221a86a58a78a56a45a127a144a139a1a199a56a45,a25a243a77a18a123a22a111a210a114
a199a77a18a234,a77a18a234a49a128,a114a199a123a56a45a10a39a243a212a174a222a107a47a53a3a21a48a234,a79a36a1
a91a176a62a7a195a85a212a174a25a54a210a137a7a1a91a99a123a150a39,a248a234a22a111a131a105,a164a139a14a145a45a123
1
a10a39,a243a5a193a123a49a165,a183a46a227a135a203a76a27a89a24a89a1a91a48a193a123a21a48,a135a232a73a164a139
a24a199a21a48,a33a7a155a123a52a133a106,a105a219a5a123,a202a47a123a106a248a52a34a106,a0a143a243a34a106a173
a222,x2 + 1 = 0 a210a150a139a137,a243a133a106a173a222,a183a162a88a98a38a199a139a137,a63a37a133a106a139a58a158
a28a201a123a117a159,a139a105a245a25a54a209a250a56a243a133a106a176a62a234,a133a106a166a34a106a24a248,a139a228a174a123
a21a70,a28a126a89a199a21a70a131a128,a133a106a83a44a234a105a245a25a54,a183a162a49a128a204a23a116a243a133a106a176
a62a123a89a74a19a142,a196a49,a106a166a7a97a126,a183a162a89a199a49a52a97a126a106a166,a7a166a204a97a126,a7
a97a126a123a155,a204a21a25a133a106a105a7a155,a79a36,a133a35a60a106a88a24319a45a22a123a21a48a52a106a166a176
a62a33a7a155a123,a52a24a199a7a217a199a48a193a123a21a48a117a122a205a7a155a24a74a123a21a48,a33a128,a183a122
a19a154a30a45a89a89a243a133a106a48a193a123a97a126,a133a106a88a144a52a38a122a233a149a127a24a199a48a199a42a139a137,
a63a37a188a126a133a106,a105a245a106a166a175a11a117a122a59a92,a133a35a60a106a88a7a34a35a60a106a88a59a92a245
a234,a34a35a60a106a88a139a142a245a228a54a123a175a11,a217a34a166a34a19a88a76a139a7a248,a88a44a104a30a15a137
a7a234,a196a49a89a52a220a199a11a248,a183a7a243a89a199a49a199a176a62a178a199a162a46a205a27a89a24a154,a38a122
a76a27a21a234a137a135a232a73a243a199a162a222a123a97a126a52a33a173a7a123a220a199a48a53.
2 a7a149Stokesa218a42a123a86a223
a222a39a89a116a234a135a232a73a123a228a253a189a174,a139a30a127a210a85a196a89a171a111a231a213
integraldisplay b
a
f(x)dx =
integraldisplay
f(x)|ba → L(
integraldisplay
f(x)dx) = f(x),(2.1)
a253a89a220a199a42a6a35a127,a105a110a99a143,a104a35a183a243a8a13a135a205a236a39a135a232a73,a183a7a3a210
a139a24a199a175a11,a143a31a143a89a210a52a228a253a189a174,a41a170a88a195,a105a88a115a143,a28a162a76a150a25a243a15
a164a139a89a199a243a14,a88a163a175a175a11,a183a0a30a15a88a163a175a175a11,a203a28a162a25a243a24a248,a41a170
a88a195,a44a234a105a245a35,a98a127a119(2.1)a123a104a52a228a253a189a174,a89a52a79a143(2.1)a128a210a234a135a73
a166a232a73a123a7a248,a89a199a42a6a123a220a224,a24a30a52a199a189a232a73,a52a24a199a193a232,a141a30a52a135
a73a35a39a123a228a174,a196a49a141a30a123a232a73a52a24a199a88a189a232a73,a166a233a155,a52a24a199a60a106,a199a123
a135a73a52f(x),a15a210a52a128,a199a123a38a30a52a232a73,a141a30a52a135a73,a0a143a89a199a228a253a189a174
a210a128a210a234a135a73a166a232a73a123a228a253a7a248,a76a151a222a128,a135a73a52a232a73a123a24a199a39a228a174,a210
a52a7a73a24a199a60a106,a38a122a199a42a143a46a127a60a106a123a135a73,a25a243a123a175a11a52,a116a234a176a145a14
a143a248a218a89a199a228a253a189a174a52a24a199a35a106a123,a25a243a34a247a245a35a106,a204a14a143a248a218a89a210a52
2
a245a35a106a123a135a232a73,a139a24a199na145a123a56a45,na145a5a117a210a139a142a245a88a51a123a145,a245a35a106a123
a135a232a73a228a253a10a39a52a199a173a232a73,a243a178a193a222a52a24a199a19a173a232a73,a243a176a145a123a56a45a52a245
a173a232a73,a183a222a39a89a234,a232a73a139a24a199a232a73a123a75a173,a232a73a52a243a24a199a75a173a176a70a232a73,
a108a128a164a139a24a199a174a6,a204a7a255a88a232a31a143a192a220,a89a24a199a60a106a52a31a143a192a220,a183a222a39
a89a89a199a174a6a52a24a199a105a135a73,a105a135a73a210a52dx,dy a89a74a135a73a198a229a117,a88a44a89a199a198
a27a52a39a233a117a123,a39a233a193a201a244a234,a79a143a39a233a193a131a128,a24a199dx a88a21a234a63a243a220a39,
a253(dx)2 = 0,a24a199a7a155a123a175a11a52a31a143a119dx,a89a199a175a11a7a118a133a236,a89a229a117a7a118
a159,a89a199a175a11a15a210a52a31a143a52a24a199a60a106a123a135a73,a183a162a34a189dxa52a104a189a123,a139a63a66
a123,a49dxa143a35a106a5a24a199a245a49a42,a89a199a245a49a42a123a198a27a52a39a233a193,a89a171a39a233a193a198
a27a123a245a49a42a119a105a135a73a42,a105a135a73a42a210a52a141a232a73a123a24a199a233a54,a243a24a199a75a173a176a232
a89a199a105a135a73,a89a15a44a49a23a42a24a171a123a83(pair),a139a24a199a75a173,a242a139a24a199a232a73a90,
a56a243a24a229,a232a73a139a24a199a138,a89a199a138a52a24a199a106,a89a220a199a52a123a92a123a136a42,a139a234a89a199
a245a173a232a73a123a10a39a131a128,a245a35a106a123a135a232a73a228a253a189a174,a210a52a196a162a123Stokesa189a174,St
okesa189a174a52a24a199a1a91a123a25a54a166a24a199a73a219a25a54a201a92a123a136a42,a243a176a145a30a127,a190a152
a243na145a123a56a45,a34a38a63a243a24a199ka145a123a75a173,a199a44a49a52a132a145a123a127a63a75a173,a139a89a248
a24a199ka145a75a173,a190a152a178a193a222a24a199a19a145a75a173,a56a45a165a24a199a77a193a127,a105a210a23a143,a89
a199a132a145a75a173a139a24a199a30a142,a75a173a139a30a142a123a10a39a52a83a106a95a192a24a199a228a253a10a39,a28a7
a207a196a199a123a30a142a7a248,a24a199a253a47a123a207a196a210a90a116a196a162a195a5a196a51a174a107(homology).
a51a174a107a52a83a106a95a192a207a196a56a45a117a159a123a33a228a253a123a24a199a10a39,a25a243a139a24a199ka145a75
a173?,a199a123a30a142a85a196,a12a105a139a24a199(k?1)a145a105a135a73,a105a135a73a42a6a52k?1a39,
a135a73a49a128a143ka39,a196a162Stokesa189a174,a210a52a128a233a149ωa52a24a199k? 1a145a105a135a73a42,a199
a123a105a135a73dω a243?a222a232a73a127a149a178ωa243?a30a142a222a70a232a73,a253
integraldisplay
dω =
integraldisplay
ω,ω a52a105a135a73a42,(2.2)
a89a199a210a52a196a162Stokesa189a174,a89a52a245a35a106a135a232a73a123a24a199a228a253a189a174,a243a215a12a11a9
a85a123a86a165,a15a1a47a10a241a89a199a10a154,a89a199a228a253a189a174a123a104a221a105a183a222a193a89a123a0a199a228
a253a189a174a42a143a1a47a60a111,a34a38a89a199a56a45a521a145a123,a2431a145a123a60a111,a75a173a52a34a227,a199
a123a30a142a210a52a34a227a123a220a199a224a154,a220a199a154,a70ωa243a220a199a224a154a123a138(a217a165ωa143a0a199a88
a189a232a73)a210a52a183a243a222a193a228a253a189a174a123a218a42a123a141a30a60a106a243ba123a138a62a86a243aa123a138,a210
3
a52a243a30a142a222ωa123a232a73,a13a38a30a210a52a243a89a199a34a227a123a232a73,a210a52a44aa116ba123a189a232a73.
a196a49,a88a10a23a241a117Stokesa189a174a243a134a34(1a145)a123a60a111a210a52a135a232a73a123a228a253a189a174,a0
a1432a145a123a60a111a17a2182a145a210a52a105a139a214a123a196a162Greena189a174:
integraldisplay
(Px?Qy)dxdy =
integraldisplay
Pdx + Qdy (2.3)
2a145a60a111a30,a75a173a522a145a123,a199a123a30a142a52a77a34,a149a52(2.3)a210a52a178a158a123Greena189
a174,a28a162a209a127a119,a209a105a89a231,a196a49Stokesa189a174a243a178a193a222a123a1a47a60a111a210
a52Greena189a174,Stokesa189a174a139a88a51a123a214a9,a23a28a126a253a24a253a86,a88a44a25a243a7
a118a47a113a119Stokesa189a174,a0a143a164a139a12a105a123a24a199a1a47a60a111,a243a174a145a56a45,a34a247a139
a24a199a77a193,a199a52a24a199a174a145a75a173a123a30a142,a0a143CIa36a30Storkesa189a174a210a85a196
integraldisplay
(Px + Qy + Rz)dxdydz =
integraldisplay
Pdydz + Qdxdz + Rdxdy,(2.4)
a28a162a243a166a176a127a135a232a73a46a178a141a116a234,a24a199a19a39a42a243a30a142(a77a193)a222a123a173a232a73a127a149
a199a123a174a39a42a243a75a173a176a62a123a174a173a232a73,a89a52Stokesa218a42a123a12a105a24a199a60a89,a114a199
a123a60a89a243a176a145a209a233,a139a24a199a228a253a117a159,a210a52a105a135a73d a126a220a39a24a189a127a1490,a34
a38?megaa52a24a199a105a135a73a42,a0a143
d(d(ω)) = 0,(2.5)
a89a199a48a199a58a158a142a52a121a210,a233a149ω,a105a135a73a42a23a108a52a34a117a123,a196a49a28a144a137a7
a178ωa104a196a24a199a92a49a117a121a210a210a113a234,a89a52a79a143a28a154a24a49a123d2a209a127a1490,a149a52a233a149
a92a49a123a60a89,a92a49a52a24a28da198a222a24a199a60a106,a23a108,a144a7a121a210a24a199a60a106a126a220a39d,
a199a24a189a127a1490a210a44a49a234,a183a142a5a174a234a24a5,a243a24a199na145a123a56a45a165,a199a123a43a41
a52(x1,···,xn),a139a24a199a60a106a52f a52xia123a60a106,da24a39a123a155,a210a52a202a47a123a160a135a73,
a15a210a52?f?xidxi,a242a135a73a24a39,a122a116a19a255a160a135a73,a242a198dxj ∧dxi,a89a199a19a126a160a135
a73fija52a233a193a123,a89a52a79a143a70a160a135a73a166a39a147a195a7,a79a36a89a199a248a106a52a233a193a123,a13
a183a162a89a220a199dxi,dxja123a198a27a52a39a233a193a123,a23a108a220a39a135a73a131a128a210a127a1490a234,a253
d(df) = d(fidxi) = fijdxj ∧dxi = 0,(2.6)
a89a176,a79a143a251a189a234ia166j,a210a122a116dfij?dfji,a98a52a79a143fa52a233a149a89a199a141a41a52a233a193
a123,a196a49a210a520a234,a79a36a222a193a121a210a234a233a149a60a106a123d2 = 0,a89a210a44a49a234,Stokesa189
4
a174a44a49a128a75a173a166a105a135a73a52a24a199a233a83,a38a122a70a30a142a203a174a89a199da89a220a199a174a6a52adj
oint,a52a233a83a123a174a6,a89a52a199a234a88a122a123a136a42,a79a143a70a30a142,a52a24a199a1a91a228a174,
a217a34a70a24a199a75a173a123a30a142a52a24a199a113a92a123a1a91a123a228a174,a52a114a199a123a75a173a123a24a199a117
a159,a70a105a135a73da52a24a199a219a92a123,a73a219a123a228a174,a52a113a92a219a92a123,a144a166a89a24a154a123a142
a163a139a7a248,a196a49a24a199a52a114a14a123a1a91a174a6,a24a199a52a219a92a123a73a219a174a6,a199a162a52
a233a83a123,Stokesa189a174a128a199a162a52a233a83a123,a196a49a89a52a24a199a173a7a244a234a123a189a174,a183a44
a49a5a193a227a135a89a122a245a24a154,a56a45a88a24a189a52a202a47a123Euclida56a45,a15a142a56a45a252xa41
a43a41a143a196a162a123a22a111,a34a247a56a45a52a24a199a22a111a123a155,a15a44a49a255a88a199a123a105a135a73a42,
a190a152ka39a123a105a135a73a42,a24a199ka39a123a105a135a73a42a28a12a105a24a199ka39a105a135a73a42a164a52a24
a199ka39a123a105a135a73a42,a149a52a196a139a123ka39a123a105a135a73a42a196a143a24a199a183a162a196a162a123a37a222a56
a45(vector space),a243a217a165a44a49a159a113a28a62,a25a243a183a210a255a88a196a139d = 0a123a89a171a105
a135a73a42,a253a105a135a73a1430a123a0a74a105a135a73a42,a243a106a166a222,a183a162a193a89a171a105a135a73a42a52a85
a20(close)a123,a89a74a85a20a123a105a135a73a42a232a196a37a222a56a45,a79a143a220a199closea105a135a73a35a28
a133a143a85a20a123,a247
Γk = {ω|ω a52k a39a105a135a73a42},Ck = {ω|ω ∈ Γk,dω = 0},(2.7)
a0a143a183a82ω,ωa52a24a199a85a20a123a105a135a73a42,a25a243a183a178Cka104a196a24a199a107,a89a199a107a139a199
a6a107,a89a199a6a107a52a31a143a17a218a199a210a52a196a139a123k? 1a145a123a105a135a73a42a6a126da117a42a126.
a79a143d2 = 0,a196a49a199a210a24a189a52closea123,a79a36a243a196a139a123a85a20a123ka39a105a135a73a42a232
a196a123Ck a165,a196a139da198a222a24a199k?1a39a105a135a73a42a196a143a24a199a6a107,a149a52a114a199a107a126a6
a107a24a248,a243a107a88a176a62a128a199a52a24a199a219a107(quotient),a89a199a107a139a199a214a9a119de Rham
group:
Hk = Ck/dΓk,(2.8)
a89a243a95a192a222a58a158a173a7,a210a52a128,a105a135a73a42a245a122a88a122a234,a4a150a149closea123a105a135a73
a42a15a245a122a88a122a234,a13a243a28a248dβa131a128,a243a105a245a60a111a131a5,a210a35a196a24a199a139a33a145a123
a37a222a56a45,a0a143a89a199a139a33a145a56a45a123a145a106a52a89a56a45a123a24a199a173a7a123a117a159,a47a158a119
a41Betti number,a89a52a83a106a95a192a165a33a6a123a24a199a228a253a10a39,a15a210a52a183a162a255a88a105
a135a73a42a44a49a251a189a199a123a139a74a95a192a123a88a35a42.
5
3 a141a106a90a233a106a60a106
a183a25a243a86a89a12a105a24a199a175a11,a222a39a139a124a89,a233a149a203a89a199a49a139a74a104a10,a183a89a123a89
a74a11a248a88a24a189a139a7a248,a196a49a28a152a42a233a236a24a199a11a248a139a104a10a123a155,a210a38a183a89a24a199
a47a123a11a248a234,a196a49a88a24a189a73a245a232a107a47,a25a243a183a166a199a11a248,a135a232a73a17a108a52a207
a196a60a106a123a117a159,a126a135a232a73a117a44a43a199a123a117a159,a0a143a60a106a52a245a122a88a122a234a123,a60a106
a139a171a171a123a117a159,a13a139a24a74a60a106,a7a118a59a92,a79a36a15a7a118a173a7a63a37a142a245a97a126a222
a15a141a116,a139a220a199a1a47a173a7a123a60a106a52a141a106a60a106(exponential function)a166a233a106a60
a106(logarithm fun ction),a89a220a199a60a106a139a31a143a117a159a17? a89a52a58a158a173a7a123a60a106,
a183a162a209a65a122a62a24a199a123a135a73a42,a13xn+1a123a135a73a52a127a149(n + 1)xn,a79a36xna123a232a73
a52a127a149 1n+1xn+1,a89a176a34a38n + 1 negationslash= 0,a253
d
dxx
n+1 = (n + 1)xn →
integraldisplay
xndx = 1n + 1xn+1,n + 1 negationslash= 0,(2.9)
a152a42na88a127a149?1,a202a47a124a116a89a199a30a127a210a136a101a234,a79a143a28a127a119a89a199a218a42a52a31a143
a30a127a196a193,a89a199a218a42a243n =?1a30a88a233,a89a210a234a234,a89a52a105a7a108a123,a88a44,a152a42
a89a30a127a7a42a142a123a155,a28a210a150a139a126a116a60a106a232a73a123a173a7a123a189a174,a79a143n =?1a30,
a89a199a232a73a98a139a63a135,a196a49,a34a38n =?1,a183a210a82a233dx/xa123a232a73,a79a143a183a88
a82x = 0,a196a49a183a89a199a232a73a34a189a199a441a232a116x,a89a199a232a73a52a7a155a244a234,a139a63a66a244
a234,a79a143a183a89a199a232a73,a119a199logx:
integraldisplay x
1
dx
x = logx,(2.10)
a89a210a52a233a106a60a106,a5a193a183a255a88a233a106a60a106a123a33a173a7a123a117a159,a34a38a183a178xa198a158
a106a,a233logaxa70a135a73,a132a149loga123a135a73a127a1491/x,a149a52logaxa15a52a127a1491/x,a196a49
a89a220a199a60a106a139a24a199a158a106C:
d
dx log(ax) =
1
axa =
1
x =? logax = logx + C,(2.11)
a34a38a183a82x = 1 a83a156(2.11),a36a30log1 = 0,a89a52a79a143a232a73a52a441a116x,a196a49
a441a1161a232a73a104a108a520,a149a52a183a210a121a116a158a106Ca210a52loga,a79a36a210a122a116loga89a199a60
a106a123a228a253a117a159:loga60a106a126a116axa123a155a127a149logx + loga:
log(ax) = loga + logx,(2.12)
6
a166a233a155a128,a233a106a52a38a122a198a27a35a143a28a27,a89a52a44a3a126a233a106a44a15a174a123a24a199a228a253
a117a159,a25a243a79a143a139a15a174a229a234,a76a150a88a76a126a234,a88a44loga89a199a60a106a58a158a7a155,a79
a143a126a116a234a183a162a89a199a228a253a123a117a159(2.12),a0a143a132a89a199a233a106a123a60a106a193a47a210a90a159
a141a106a123a60a106,a141a106a60a106a52a233a106a60a106a123a39a60a106,a34a38y = logxa123a155,a210a149a189a66,
x = ey,a253
y = logx? x = ey,(2.13)
a79a36a199a162a144a35a52a35a39a123a60a106,eya52a199a141a106a60a106,a217a166logxa24a229a139a28a27a166a198a27
a7a248a123a218a42,logxa178a198a27a35a143a28a27,a141a106a60a106a15a210a178a28a27a35a143a198a27a234,a24a199
a178a198a27a35a143a28a27,a24a199a112a234a44a117,a199a210a178a28a27a35a143a198a27,a89a52a24a199a59a92a123a218
a42:
ex+y = exey,(2.14)
a183a89a176a139a199a121a210:
ex+y = elogu+logv = eloguv = uv,u = ex,v = ey,(2.15)
a89a74a209a52a105a142a52a123a15a174,a183a25a243a7a121a210a141a106a60a106a199a123a135a73a210a52a199a7a3,
a253eya233ya70a135a73a210a127a149ey,a89a199a121a210a143
dy = dxx? expydy = dxdy = x = expy,(2.16)
a89a176a178a141a106a60a106a85a196expy,a104a108a15a44a85a196ey,a89a30a34a189a196a139a123a106a209a52a116
a123,a196a49a150a139a31a143loga63a243a166a100a123a175a11,a149a52,a141a106a60a106a123a135a73a210a52a199a7a3,
a222a193a210a201a241a234a24a199a121a210,a76a27a15a142a16a122,a220a199a60a106a123a67a52a89a199a248a6,a24a199
a52loga243xa123a75a173a165a243x = 0a123a142a163a214a117a214a66a229a117,a74a149a139a195a65,a233a106a60a106a15a52
a24a199a15a159a123a60a106(increasing funct ion),a88a44a199a15a159a122a58a158a122,a141a106a60a106a210a15
a159a122a105a79,a199a123a207a52a116a123,a89a52a183a152a123a220a199a59a92a123a67(graph).a183a46a28a162a243a127
a91a135a232a73a123a86a209a23a116a44a89a220a199a60a106a123a67,a141a106a60a106a166a233a106a60a106a52a58a24a123a60
a106,a24a199a52a12a105a24a199a123a39a60a106,a89a199a117a159a52a58a158a7a155a123,a139a219a201a123a117a159,a145
a24,a141a106a60a106a123a135a73a52a199a7a3,a79a36,a199a139a24a199a105a59a92a123a195a65a255a106,a89a199a195a65
a255a106a52a126Talyora218a42a48a13a123,a183a178Taylora218a42a85a24a5a213
f(b) = f(a) + f
prime(a)
1! (b?a) +
fprimeprime(a)
2! (b?a)
2 +···+ Rn,(2.17)
7
Rn =
integraldisplay b
a
f(n)(t)(b?t)
n?1
(n?1)! dt,(2.18)
a183a46a89a74a28a162a209a39a44a234a89a74a234,a89a199Taylora218a42a178a127a63a123a60a106a48a196a24a199a195a65
a255a106,aa52a24a154,ba52a12a105a24a154,a0a143a199a44a49a48a196(b?a)a123a24a199a245a49a42,a128a193a139
a24a199a155a49,a89a199a155a49a52a132a232a73(2.18)a201a241,Taylora218a42a243a24a199a155a49a123a30a127a210
a52a89a199a196a162a165a138a189a174a195Mean Value Theorema196,Taylora218a42a210a52a165a138a189a174a123
a176a39a123a24a199a77a18,a132Taylora218a42,a25a243a183a162a89a199a141a106a60a106a59a92a122a88a122a234,a79
a143a135a73a5a86a209a52a199a7a3,a196a49a139a24a199a195a65a255a106,a105a59a92,a183a85a143
ex = 1 + x + 12!x2 +···+ 1n!xn +···,(2.19)
a196a49a89a199a141a106a60a106a139a24a199a105a59a92a123a48a13,a199a139a24a199a173a7a123a117a159,a183a46a139a30a127,
a28a89a199a106a248a104a108a52a116a106,a150a232a52a34a106,a139a123a30a127a28a126a24a5a133a106a123a155,a139a105a28
a201a123a117a159a188a51a248a123,a183a127a119sinxa166cosxa139a12a105a89a220a199a48a13:
sinx = x? 13!x3 + 15!x5?···; (2.20)
cosx = 1? 12!x2 + 14!x4?···,(2.21)
a28a204a21a25,a34a38a233ex,a82xa149a143ix,a217a165i2 =?1,a0a143eixa210a139a199a42a6:
eix = cosx + isinx,(2.22)
a183a46a105a142a52a132(2.19)-(2.21)a23a241a117a139a89a248a123a218a42,a227a142a35a106a82a133a106a123a138,a82
a24a199a192a220,a0a143a34a38a28a126a89a199a218a42a123a155,a82x = pi,a149a52eipi =?1,a28a162a209a127
a119a89a199a218a42,a88a44a89a52a199a105a139a63a135a123a42a6,a79a143a89a176a62a139a1a199a158a106,a76a27
a213a63a123a24a199a52pi,a12a105a24a199a210a52e,ea52a79a143Euler,Eulera243a104a3018a45a22a123a0a199
a30a127,a0a30a203a25a243a30a45a88a212a24a248,a0a199a30a127a45a142a210a52a220a78,a45a142a139a41a166a123a21
a48,a210a52a243a220a78,a76a27a203a128a139a24a199a33a149a76a123a106a166a27,Eulera52a0a30a250a203a128a33a149
a76a123a106a166a27,a196a49a139a124a41a234a41a124a131a128,a243a198a123a170a41a176a209a63a139a199a149a76a123a106a166
a27,a149a52Eulera210a250a62a116a18a10a122a227,a198a210a85a234a105a245a105a245a86,a89a199Eulera52a105a139
a63a135a123,a76a150a85a123a169a57a52a150a139a124a167a44a123,a198a85a234a1a186a253,a198a139a80a245a66a47,a196
8
a49a198a52a230a248a66a47a6,a47a6a43a243a198a79a222,a41a198a123a106a166,a183a203a198a17a178a21a9a24a199a7
a248,a210a52a183a162a89a199a8a13a67a86a12a7a88a7a198a123a92a248,a198a123a92a248a7a1a250a76a155a151,a1
a186a253,a105a230a11a123,a128a117a183a251a189a88a111a234,a212a34a234a63a37a57a89a150a139a124a23a234,a169a57a209
a52a110a182a169a123,a136a42a183a162a67a86a12a150a139Eulera123a92a248,a183a162a139a105a245a217a199a124a123a92a248,
a88a44a25a243a23a117a175a11a52a139a74a169a9a123a175a11,a7a48a128,a28a162a152a42a139a213a101a123a155,a44a49
a23a23a176a132a123a92a248,Gaussa166a98a183a128a5219a45a22a33a149a76a123a106a166a27,a76a27a209a127a119a198
a123a106a166a21a197,a196a49a128a17,a152a42a124a27a44a129a62a198,a81a248a198a24a229a86a44a129,a0a143a44a129a30
a210a175a198a66a123a1a91a175a11,Gaussa104a108a210a21a137,a0a143Gaussa123a66a123a1a91a175a11a123a137a243
a198a123a88a169a248a176,a89a52a105a139a63a135a123,a89a74a175a11a113a92a52a240a127a1a91a123,a139a123a210a24a16.
a41a41a198a123a66a123a1a91a189a174,a198a123a66a189a174a209a105a139a63a135,a183a162a44a49a60a198a123a117a85a24a159
a169a57,a105a88a115a143a52a198a123a169a57a88a52a110a182a169,a210a52a121a169,a150a232a52a121a169,Eulera85a234
a105a245a192a220,a89a199a218a42a52a217a165a131a24,a199a178a1a199a204a7a123a106a203a229a117,e,i,pi a139a89a248
a123a7a248,eipi =?1,a89a52a58a158a139a63a135a123,a183a242a183a170a28a162a12a105a24a199a218a42,a88a127
a28a162a139a195a108a85,a51a248a126a89a199a48a13a123a155,a44a49a126a116arctgx,arctgxa123a135a73a52 11+x2.
a196a49arctga89a199a60a106a139a24a199a117a159,a210a52a24a134a70a199a162a135a73a123a155,a42a6a204a7a118a59a92.
a79a36a132a89a74a210a122pia123a24a199a42a6,a253pia44a49a85a196a24a199a195a65a255a106
pi
4 = 1?
1
3 +
1
5?
1
7 +··· (2.23)
a89a52a24a199a164a224a244a234a123a24a199a218a42,a199a178a24a1991,3,5,7,9a89a143a203a243a24a229,a28a243
a24a229a52pi/4 a188a196a49,a89a52a164a224a244a234a123a24a199a218a42a188a163a83a123a24a199a139a214a123a106a88
a27Selberg,a198a52a73a133a123a106a166a27,a139a24a39a198a85a24a199a169a57,a89a198a233a106a88a21a9a108a85
a210a79a143a198a23a116a89a199a218a42,a198a57a89a25a243a52a181a106a76a234a24a154,a183a46a198a44a49a15a13980a245
a181,a124a164a243.
4 a243a1a91a222a97a126
a164a139a1a73a168,a5a24a39a183a7a89a24a154a1a91,a135a232a73a243a1a91a222a123a97a126,a1a91a222a123a97
a126,a104a108a52a56a45a1a91a33a139a63a135a123a52a77a193a123a1a91a9a9a19a145a77a193a123a1a91,a19a145a77
a193a139a171a171a248a6a199a139a171a171a123a111a42a199a139a171a171a88a51a123a117a159a2a243a89a48a193a139Gaussa123
9
a211a42,a210a52
ds2 = dx2+dy2+dz2 = Edu2+2Fdudv+Gdv2,x = x(u,v),y = y(u,v),z = z(u,v)
(2.23)
a89a199a77a193a47a158a126a107a106a44a43a199a243a89a199a174a145a56a45a165,a43a41a52(x,y,z),a149a52a178
a174a145a56a45a123a43a41a44a43a143a220a199a35a106a123a60a106,a196a49x,y,za52a220a199a35a106u,va123a60
a106a199a183a162a193a131a143a107a106(par ameter).a149a52a89a199a56a45a176a62a139a24a199a225a229a172ds2a199a210
a52dx2 + dy2 + dz2,a178x,y,z a44a43u,va123a60a106a131a128a199ds2 a210a35a143a24a199a19a39a123a135
a73a42,a217a165a248a106,a24a196a210a119a41E,F,G,a183a162a65a122a1a91a13a41a123a30a127a139Euclida1
a91a199Euclida1a91a105a149a76a188a199a52a62a24a253a114a199a123a1a91,a199a23a241a117a7a139a24a171a218a174a122
a116a106a166a123a136a88a52a105a173a7a123a199a79a143a106a166a136a88a52a132a24a199a218a174a178a44a24a199a92a246a123a77
a174a122a117,a79a36,a89a199a52a105a228a14a105a42a251a123a24a199a136a88,a13a37a199a217a34a88a144a52a1a91,
Euclida89a199a21a1a91a198a253a22a52a114a199a123a106a166,a198a23a241a117a132a218a174a126a92a246a48a27a77a241a136
a88a123a173a7a117,a233a149a89a48a193a183a250a122a105a110a99a123a52a199a165a41a150a139,a183a162a89a199a49a52a97a126
a106a166,a233a149a97a126a106a166,a165a41a212a213a173a97a126a234,a127a91a192a220a209a24a189a7a139a97a126,a13a233
a149a89a248a123a24a199a106a166a76a27a49a143a150a139a31a143a97a126,a217a34a33a240a28a7a123a52a33a240a41a234a24
a154a123a155,a97a126a204a117a123,a63a37a97a126a205a173a7,a205a253a47,a233a149a89a199a19a39a135a73a42(2.23),
Gaussa123a211a42a210a52a44a49a202a226a89a199a19a39a135a73a42,a21a48a24a199a1a91,a89a199a1a91a210a76
a122a88a122a234,Euclida1a91a44a49a199a123a135a73a42a210a52du2 + dv2,a89a52a24a199a33a59a92a123a60
a111,a12a105a24a199a60a89a131a5a210a44a49a21a48a58a78a1a91,a25a243E,F,Ga52a127a63a60a106,a242a28
a222a24a74a55a104a30a71,a89a199a1a91a123a10a39a18a76a122a88a122a234,a210a52a128,a243a174a145a56a45a44a49
a139a35a193,a253a35a193a222a123a1a91a199a44a49a88a11a89a199a77a193a243a174a145a56a45a176a123a160a152,a144a31
a68a243a77a193a222a123a89a199a1a91,a199a221a105Euclida1a58Euclida1a91a127a243a19a199a18a122a234a88a122.
a149a52,a89a248a123a24a199a21a25a38a122a1a91a139a105a245a199a78a60a1a91a123a10a39a15a210a77a18a116a24a196
a123a60a89,a196a49,a183a5a39a7a89a154a135a232a73a243a1a91a222a123a97a126,a183a46a152a21a15a142a139a88a245,
a89a113a234,a92a92a188
10
a145a174a89 a77a34a88
2001a3510a21926a134
1 a178a193a77a34
a183a46a89a1a39a203a76a27a89a24a154a135a232a73a243a1a91a222a123a97a126,a89a52a58a158a7a155a123a21a48.
a0a143,a44a33a59a92a123a60a89a13a41,a183a162a210a89a178a193a222a123a77a34,a34a247a178a193a222a139a24a30a77
a34x(t) = (x1(t),x2(t)),a253a243a89a199a67a222a196a243a123a60a89,a126a135a232a73a123a155a17,a210a52a89
a30a77a34a139a30a35a34,a35a34a139a199a35a37a222,a233a149a35a37a222,a183a162a82a89a199a37a222a52a92a160a37
a222,a199a123a159a221a521,a15a210a52a82a143a92a160a35a37a222,a149a52a183a162a127a119a34a38a178a43a41xa44a43
a196a140a159sa123a60a106a123a155,a89a210a44a43a89a199a92a160a35a37a222a210a52x a233s a123a135a73dxds,a253a92a160
a35a37a222a143
e1 = (dx1ds,dx2ds ),(e1,e1) = 1,sa52a140a159,eqno(3.1)
a0a143a14a143a248a207a196a89a30a35a34a17a218a105a59a92,a0a210a52a139a234a24a199a92a160a35a37a222a131a128,a63
a34a247a152a42a178a193a52a189a53a123,a253a139a24a199a221a196a123a48a53,a0a143a199a210a139a24a199a92a160a27a53a222,
a15a210a52a203a199a18a134a123a0a199a92a160a37a222,a25a243,a183a119e1a52a92a160a35a37a222,e2 a52a92a160a27
a37a222,a149a52a7a122a116a89a30a35a34a123a117a159,a145a24a71a47a60a210a52a178e1a89a199a60a106a233a149sa242a70
a135a73,a0a143a242a70a135a73a131a128,a104a108a89a52a24a199a99a123a37a222,a79a143e1a52a24a199a92a160a37a222,
a196a49(e1,e1) = 1,a0a143a178a199a135a73a24a5a6,a183a162a210a122a116de1ds a51e1a18a134,a196a49a199a24
a189a243a27a34a123a48a53,a79a36,a183a162a210a139de1ds a127a149a92a160a27a37a222e2a123a245a106,a89a199a245a106a52
a140a159a123a24a199a60a106,a183a162a119k(s),a89a199a245a106a119a22
de1
ds = ke2,e2 a52a92a160a27a37a222,(e1,e2) = 0,(3.2)
ka89a199a60a106a24a196a119a41a77a71,a52a89a30a77a34a243a89a199a178a193a176a62a33a7a155a123a24a199a117a159,
a52a140a159a123a24a199a60a106.
a243a11:
a233a149a201a189a123a77a34a48a199,a201a241a77a71k a123a218a42,[a10a43a213ka52a77a34a48a199a123a24a126a90
a19a126a135a73a123a24a199a60a106]
1
2 a56a45a77a34
a44a178a193a77a34a242a159a24a90a14a143a248a17a218a183a162a23a23a56a45a123a60a111,a34a247a183a162a25a243a139a24
a30a77a34a52a56a45a123a77a34x(t) = (x1(t),x2(t),x3(t)),a2433a145a56a45a176a139a35a34,a196a49
a89a199a56a45a123a43a41x,y a210a44a43a143a107a106t a123a60a106,a89a74a192a220a28a162a76a150a209a127a119,a183
a242a167a243a24a5a6,a196a49x(t) a52a24a199a37a222,a199a123a73a222a210a52a154a123a43a41,a13a154a52t a123
a60a106,a149a52a199a210a42a24a30a35a34,a893a199a73a222a119a41xi(t)(i = 1,2,3),x(t)a52a24a199a37
a222,a52a107a106ta123a60a106,a199a1233a199a73a222a210a52a178a154a123a43a41a44a43a143t a123a60a106,a0a143a14a143
a159a113a17a218a51a248a123a48a27a210a52a233a89a30a77a34a126a135a232a73,a34a189a77a34a139a35a34,a63a37a243a35
a34a222a193a139a92a160a37a222,a253a139a92a160a35a37a222,a233a77a34a139a199a48a53,a24a134a89a143a18,a215a248a24
a199a48a53,a7a152a128a107a106t a52a15a28a123a24a199a48a53,a15a13a211a131,a183a162a210a139a24a199a92a160a35
a37a222,a119a199e1,a0a143a203a178a193a51a248a123a60a89,a178e1a89a199a60a106a233s a242a70a135a73,a210a127a149
a128a233s a70a19a126a123a135a73,a79a143e1 a52a92a160a37a222,a196a49a122a116a123a89a199a135a73a203e1 a52a18a134
a123,a13a233a149a24a30a56a45a165a123a77a34,a199a123a35a34a52a24a30,a199a123a27a34a139a195a106a199,a217a34
a178a44a89a24a154a203a35a34a18a134a123a27a34a139a195a106a199,a0a143a217a165a139a24a199a52de1ds,a199a52a88a113a92
a104a189a123,a132a149a51a248a123a174a132,a183a178a89a199a48a53a123a92a160a37a222a119a41e2,a0a143a199a210a52ke2,
a253a44a49a85a196 de
1
ds = ke2,(3.3)
e2a52a217a165a24a30a27a34,a183a162a119a199a143a204a27a34(principal normal),a13ka89a199a60a106a166a178
a193a123a60a111a24a248,a119a41a77a34a123a77a71,a196a49a25a243,a183a139a24a199a35a34a123a48a53a90a24a199a204
a27a34a48a53,a243a174a145a56a45a210a139a12a105a24a199a48a53a203a89a220a199a48a53a18a134,a183a162a47a158a119
a199binormal,a15a13a211a131,a63a243a12a105a24a199a27a34,a196a49a199a139a220a199a27a34,a0a143a89a220a199
a27a34a196a196a123a178a193a210a52a27a178a193,a77a34a521a145a123,a35a34a521a145a123,a196a49a27a34a522a145a123,
a52a199a178a193,a0a143a139a24a199e3 = e1 ×e2,a217a165e1 ×e2 a52a37a222a232,a233a149a220a199a48a53a123
a155,a139a24a199a104a189a123a145a174a199a48a53a52a203a199a162a18a134a123,a63a37a38a122e1,e2,e3a196a24a199a116
a98a248a58,a183a162a34a189a89a199a56a45a52a189a53a123,a141a67,a38a67a209a139a24a199a104a189a123a48a53,a47
a158a183a162a209a126a141a67,a196a49a210a139a24a199a104a189a123e3a119a22
(ei,ej) = δij,(3.4)
2
a79a36a243a207a196a174a145a1a91a123a30a127,a89a248a123a174a199a144a35a18a134a123a92a160a37a222a196a196a123a67a111
a58a158a7a155,a89a52a143a31a143a17a218a89a248a24a199a192a220a183a119a199a41a37,a28a178a24a199a41a37a194a116a12
a24a199a41a37a123a228a196a52a113a92a104a189a123,a0a143a174a145a56a45a33a7a155a123a117a159a210a52a174a145a56a45
a123a228a196,a183a162a7a207a196a123a1a91a117a159a52a178a44a228a196a88a35a123,a196a49a210a7a127a119a31a143a30a127
a28a44a49a178a89a199a192a220a194a116a12a105a24a199a160a152,a31a143a30a127a199a123a160a152a35a139a243a149a24a199a228
a196,a13a41a37a210a52a89a199a228a196a137a219a123a44a43a123a48a27,a28a7a21a234a194a44a86a210a44a43a89a199a41
a37a194a116a12a105a24a199a41a37a123a228a196a52a113a92a104a189a123,a23a108,a144a139a24a199a228a196a63a37a24a189
a139a24a199a228a196a178a24a199a41a37a35a143a12a105a24a199a41a37,a79a143a7a207a196a56a45a178a44a228a196a88a35
a123a117a159,a196a49a137a219a123a48a27a210a52a188a126a41a37,a0a143a34a38a183a25a243a139a24a30a77a34,a183a88
a144a139a24a199a41a37,a89a74a41a37a52a30a45a123a60a106,a243a0a176a228a196,a79a36eia89a174a199a42a143a41
a37a123a37a222a209a52a30a45t a123a60a106,a149a52a183a44a49a70a199a123a135a73deidt,deidt a52a199a37a222,a79a143
a52(e1,e2,e3)a52a199a41a37,a196a49a127a91a24a199a37a222a210a44a49a85a196e1,e2,e3a893a199a37a222a123
a34a117a28a92,a0a143a183a178a199a227a135a77a18a24a154,a210a178a199a85a196deia127a149eja123a34a117a28a92:
dei = ωijek,(3.5)
a89a199a28a92a123a248a106a52a24a39a135a73a42,a89a52a79a143a183a25a243a41a234a24a5a135a73,a132a89a60a106a34
a122a116a24a39a135a73a42,a0a143a89a248a123a24a39a135a73a42a183a119a199ωij,a89a210a44a43a220a199a35a249a123
a41a37a123a7a248,a28a139a24a199ei a123a41a37,a107a30a139a199a35a249a123a41a37ei + dei,a0a143dei a44
a143eia123a60a106a123a30a127,a199a123a28a92a123a248a106a210a52a24a39a135a73a42,ωij a52a24a39a135a73a42,a243
a89a176,a183a123a141a41i,ja209a52a441 a1163,3 a52a183a162a56a45a123a145a106,a196a49ωij a24a225a1399a199:
i,j a154a24a199a441a1163,a196a49a24a225a1399a199.a899a199a24a39a135a73a42a52a139a7a248a123,a199a88a52a113
a92a127a63a123,a89a199a7a248a210a52
ωij + ωji = 0,(3.6)
a196a49a34a38a28a178ωij a23a196a24a1993 × 3 a123a48a106a123a195a163a123a155,a89a199a48a106a52a39a233a193a123.
a89a24a28a48a199a105a142a52a44ei,ej a123a19a232a127a149δija122a116,a178a89a199a7a248(a48a199(3.4))a135a73
a123a155,a210a193a47a122a116a89a24a28a117a159,a89a210a52a128(ωij) a52a24a1993 × 3 a48a106a123a195a163,a89
a199a48a106a123a195a163a209a52a24a39a135a73a42,a63a37a89a199a48a106a114a199a52a39a233a193a123,a166a233a155a128,
a89a199a48a106a204a110a34a222a123a195a163a520,a217a199a17,a132a149a39a233a193,a139ω12 =?ω21a127,a52a39
a233a193a123,a79a36a183a162a25a243a139a24a199a92a160a35a37a222,a139a24a199a204a27a37a222,a164a139a24a199a166a199
3
a162a18a134a123a196a24a199a41a37a123e3 = e1 × e2(a37a222a232),a233a149a89a199a41a37,a183a178a199a123a174a199
a48a199a209a233a140a159a70a60a106( a135a73),a210a44a49a178a89a199a60a106a44a143e1,e2,e3 a123a24a199a34a117a28
a92,a89a199a28a92a52a89a248a123a24a28a48a199a213a145a24a199a48a199a52de1 = ke2,a79a143a183a162a123a48a106
a52a39a233a193a123,a204a233a110a34a222a123a195a163a209a520,a196a49ω13 = 0,a98a52a217a198a123a195a163a7a213
a63e3a123a160a152,a132a149e1,e2 a160a10a123a7a248,ω13 a52a127a1490a123,a79a36ω31 a15a127a1490,a196a49
a89a199a48a106a85a241a117,a210a52a183a5a193a1233a199a48a199:
de1
ds = ke2
de2
ds =?ke1 + ωe3 (3.7)
de3
ds =?ωe2
a89a28a48a199a52a104a35a77a34a88a21a48a123a30a127a33a0a123a24a28a48a199,a183a162a47a158a119a199Frenet
a48a199,Fren eta52a27a41a123a24a199a106a166a27,a183a46a89a52a198a123a123a70a44a88a169,a198a88a24a189a52
a62a24a199a201a241a89a199a48a199,a104a30a139a1a199a124a41a241a89a199a211a42,a44a183a89a123,a28a162a44a49a23
a241a117a122a89a199a48a199a63a88a212a104a10,a79a36a183a248a234a77a71a49a105,a164a139a12a24a199a60a106ω,ω
a210a52a48a199a165a123a12a71(torison),a15a52a140a159a123a60a106,a52a44a43a56a45a123a77a34a243a228a196a107
a5a123a117a159,a196a49a56a45a77a34a139a220a199a60a106,a24a199a52a77a71,a12a24a199a52a12a71,a12a71a210a52
a195a85a199a14a143a248a88a52a24a30a178a193a77a34,a199a52a243a56a45a107a77a123a24a199a222,a196a49a56a45a77a34
a52a126a220a199a60a106a117a195a85a123,a199a162a137a219a143a195a85a89a56a45a123a117a159,a89a220a199a60a106a23a108a105
a173a7,a79a143a199a162a7a52a127a1490a123a30a127,a210a44a43a234a77a34a105a59a92a123a117a159,a7a52k = 0
a123a155,a89a77a34a210a196a143a134a34,a89a105a142a52a121a210,a183a5a193a201a241a121a210a213a79a143k = 0,a196
a49de1 = 0,a79a36a92a160a35a37a222e1a52a24a199a158a106,a79a143a89a248a199a123a135a73a98a127a1490,a0a143
a28a178a89a199a158a106e1 = Ca83a156a116e1 = dxdsa165,a242a178a199a232a73a24a5a6a210a122a116xa52s a123a24
a39a60a106a213
x = Cs,(3.8)
a196a49a89a210a52a24a30a134a34,a39a44a117a15a105a142a52a21a121a210a152a42a28a139a24a30a134a34a123a155,a199a123
a77a71k a52a127a1490a123,a79a36k = 0 a83a44a77a34a123a33a59a92a123a117a159,a210a52a134a34,a0a143a7
a213a63a123a52a243a189a12a71a123a30a127,a24a189a7k negationslash= 0,a165a52k = 0 a123,a149a52de1ds = 0,a15a210
a52a134a34a234,a89a30a199a210a150a139a27a6a189a204a27a34,a24a30a134a34a203a199a18a134a123a52a24a199a178a193,
4
a89a199a178a193a176a62a196a139a203a36a134a34a18a134a123a48a53a209a52a139a51a248a123a117a159,a196a49a210a150a139a204
a27a34,a79a36a15a88a21a189a12a71,a12a71a24a189a7a243a89a154a123a77a71k negationslash= 0a123a30a127a98a139a63a66.
a13a104a12a71a127a1490a123a30a127,a104a108a210a52a44a43a89a30a77a34a52a243a178a193a222a24a30a77a34,a5a193
a183a15a201a234a24a199a59a92a123a121a210,a79a143a12a71a89a199a60a106a52a243Frenet a218a42a123a145a174a199a218
a42a176,a196a49a132ω = 0 a44a127a36a30e3a52a199a158a106a123a37a222,a233a149e3 a89a199a158a106a37a222,a132
a149e3a52a24a199a27a34,a63a37a79a143a27a34a203a35a34a52a123a207a18a134a123a199a196a49e3 a203dxds a123a19a232
a52a123a207a127a1490,a79a36e3a7a52a127a149a158a106a123a155,a183a210a44a49a178a48a199
(e3,dxds) = 0 (3.9)
a232a73,a79a143e3a52a24a199a158a106a199a196a49a89a232a73a210a52e3 a203x a123a19a232a127a149a24a199a158a106,a79
a36a199a52a24a199a178a193a77a34,a149a52ω = 0a52a44a43a77a34a52a199a178a193a77a34,a39a44a117,a44a49
a105a142a52a121a210a178a193a77a34a123a12a71a520,a196a49a77a71a90a12a71a220a199a60a106a209a139a59a92a123a1a91
a117a159,a12a105a24a171a105a139a63a135a123a77a34a52ka166ωa209a88a1430,a98a209a52a158a106,a0a143a243a89a199
a60a111a131a5,a44a49a121a210a77a34a52a199a90a34,a210a52a89a248a59a92a123a135a232a73a123a97a126a243a9a212
a154a166a222a139a173a7a123a63a66,a79a143a183a162a127a119a9a212a154a166a123a24a199a204a7a123a154a92a212a52DNA.
DNAa52a220a30a90a34,a52a199a118a90a34,a89a52a9a212a222a58a158a228a253a131a25a54,a243a9a212a222,a89
a248a123a77a34a210a115a241a117a234,a79a36,a77a34a88a243a9a212a154a166a139a173a7a123a97a126,a15a210a52a76a27
a7a127a119a77a34a123a117a159a152a91a107a47DNAa123a154a166a117a159,a196a49a106a166a210a105a173a7a234,a77a34
a123a117a159a107a47a116a154a166a123a117a159,a131a217a52a243a154a166a176a62,a139a30a127a28a178DNAa35a228a234,a199
a123a117a159a210a149a35a234,a196a49a35a228a131a128,a106a166a117a159a210a21a9a149a35a199a199a123a154a166a117a159a15a149
a35,a255a88a199a162a152a91a149a35a199a89a52a243DNAa123a207a196a249a243a9a212a154a166a48a193a52a58a158a228a253
a123a175a11,a76a27a41a234a105a245a123a211a42,a25a243a252a24a253a135a9a212a154a166a123a86a7a32a13a117a123a155,
a210a23a68a139a24a199a228a253a123a218a42,a119a41Whitea218a42,Whitea52a183a123a24a199a166a9a123a166a9,
a198a41a89a199a211a42a52a198a70a44a88a169a123a24a199a136a42,a198a228a237a105a80,a198a89a199a136a42a35a143a9a212
a154a166a123a24a199a228a253a123a218a42,a183a25a243a88a21a89a89a199a136a42.
5
3 Feuchlla88a127a42,Croftona232a73a218a42a90Fary-Milnor
a189a174
a41a89a199a1a91a207a196a123a30a127,a139a74a173a7a117a159a7a255a88,a47a158a173a7a123a117a159a128a128a52a114a14
a1a91a123a117a159,a243a77a34a123a60a89a15a88a52a207a196a24a66a227a123a1a91,a183a162a25a243a178a222a193a123a89
a199a119a41a219a92a123a1a91a199a253a207a196a24a66a227a123a1a91a117a159,a89a30k,ωa209a52a140a159a123a60a106,
a126a199a162a207a196a199a123a117a159,a52a14a143a248a107a77,a88a44a205a173a7a123a52a207a196a114a30a77a34,a34a38
a139a24a30a77a34a52a85a20a123,a23a89a30a77a34a139a31a143a248a123a117a159,a89a176a139a199a58a158a7a155a123a218
a42,a233a149a89a30a85a20a123a77a34,a154a154a139a199a77a71,a79a143a85a20a234,a183a210a207a196a92a77a71,a253
a178a89a199a77a71a215a114a30a77a34a70a199a123a232a73,a149a52a210a139a196a162Feuchlla218a42
integraldisplay
c
kds ≥ 2pi,(3.10)
a210a52a128a89a199a232a73a139a199a5a142,a24a189a522pi,a89a199a134a250a89a229a117a210a52a152a42a28a24a154a77
a71a209a150a139,a89a30a77a34a210a134a34a18a5a86,a52a88a204a85a20a229a117,a28a7a134a34a18a5a86a21a234
a85a20a123a155,a15a7a178a199a107a44a117,a107a44a117a210a7a139a77a71,a0a143a152a42a21a234a107a44a117,a166a198
a117a143a48a233a222a117a123a155,a0a143a77a71a243a114a199a77a34a222a123a232a73a139a24a199a251a189a123a5a142,a89a199
a5a142a522pi,a89a210a52a196a162a123Feuchlla218a42,a183a7a121a210a89a199a218a42,a143a31a143a89a92a77a71
a7a139a24a199a5a142? a121a210a89a199a218a42a123a48a27a210a52a255a88a89a30a77a34a123a92a160a35a37a222,a253
a243a35a34a123a48a53a123a24a199a92a160a37a222,a204a7a10a39a210a52a188a126a176a132a110a243(Gau ss map).
a233a149a89a248a24a199a92a160a35a37a222,a243a251a189a154a82a203a89a199a35a37a222a52a178a113a123a0a74a92a160a37
a222,a196a49a28a139a142a245a92a160a37a222a52e1(s),a199a52a159a127a1491 a123a24a199a37a222,a196a49a28a34a38
a178a199a23a196a56a45a24a199a154,a199a210a52a243a92a160a69a222a123a24a199a154,a79a36a28a82a24a199a92a160a69,
a82e1(s)a23a196a92a160a69a222a123a24a199a154,a0a143a104a28a89a199a154a215a198a117a77a34a18a24a87a123a30a127,
e1(s)a243a92a160a69a222a196a24a30a77a34,a89a52a176a132a110a243a123a63a135,a253a196a162a123a176a132a110a243,a89
a30a77a34e1(s)a243a92a160a69a222a196a196a123a123a77a34a88a52a198a117a123a77a34,a166a234a24a171a234,a79a143a199
a123a48a199a52e1(s),a88a52x(s),e1(s) a119a22:
dσ2
ds2 = (
de1
ds,
de1
ds ) = k
2,σa52e1(s)a123a140a159,(3.11)
a92a160a69a193a152a42a139a24a199a76a203,a0a143a89a199a76a203a139a24a199a244a154,a183a119a199y,a15a210a52a128,
6
a76a203a52a6a204a203,y a52a89a199a76a203a123a244a154.a178a89a199a244a154a104a196a186a222a123a24a154,a0a143a77a34
a210a139a199a176a221,a89a199a176a221a250a82a196a52y(x),a52a199a19a232,a89a199a176a221a44a49a23a196a52a77a34
a222a123a24a199a60a106,a0a143a77a34a243a56a45a176a62a210a139a234a24a199a176a221,a183a162a89a77a34a52a85a20a123,
a104a108a139a199a33a176a154,a139a199a33a132a154,a243a33a176a154,a79a143a199a52a33a176a123,a23a108a199a123a35a34a52
a178a123,a243a33a132a154,a35a34a15a52a178a123,a196a49a210a139a220a199a248a142a154,a24a199a52a83a44a176a221a33
a176a123,a24a199a52a33a132a123,a13a166ya89a199a154a18a134a123a0a199a76a203,a24a189a203a92a160a69a222a123a77a34
a35a98,a79a143(y,dx) a89a199a19a232a520,a196a49ya203a69a193a123e1(s)a243a24a154a35a98,a79a36a243a92
a160a69a222a123a77a34a203a89a199a92a160a203a35a98,a98a52ya52a127a63a48a53,a196a49a183a162a69a222a123a176a132
a110a243a123a77a34a52a203a69a222a123a31a143a203a209a35a98,a89a52a79a143ya52a127a63a24a199a154,a149a52ya123a0
a199a6a204a203a98a183a162a89a30a77a34,a183a46a176a132a77a34a183a119a199γ,a199a203ya123a6a204a203a35a98,a79
a36a203a127a63a92a160a203a35a98,a89a52a105a7a155a123a24a199a117a159,a196a49γa89a30a77a34a24a189a7a35a104
a159,a199a243a69a222a88a52a24a30a127a91a123a77a34,a199a243a69a222a52a24a30a203a196a139a123a203a177(a76a203)a209
a35a98a123a77a34,a0a143a183a164a52a137a7a121a210a199a1392pi a89a199a5a142,a5a193a117a128a210a89a199a175a11.
a183a89a30a77a34γa52e1(s),a119a51a160a203,e1(s) a210a52a56a45a77a34a123a24a199a176a132a110a243,a199a52
a35a34a123a48a53a243a92a160a69a222a196a196a123a77a34,a13a89a30a77a34a119a41γ,a199a203a69a222a123a127a91a92
a160a203a209a35a98,a0a143a24a196a143,a89a30a77a34a35a104a159a234,a152a42a212a225,a199a150a139a89a199a117a159.
a128a199a139a24a1992pi a123a5a142,a52Croftona218a42a213
integraldisplay integraldisplay
= 4L,(3.12)
Croftona123a169a57a15a105a139a63a135,a88a52a24a199a116a42a123a106a166a169a57,a93a41a123a186a41a92
a86(Encyclopaedi a)a62a198a85a24a159a169a57,a52a7a149a1a91a150a71a123,a198a52a243a85(a1a91)a150
a71a123a169a57a30a127a178a198a123a136a42a85a243a176a62a234,a89a199a136a42a105a7a155,a166a233a155a128,a210a243a178
a193a222a89,a77a34a123a159a221a44a49a44a143a203a199a35a98a123a134a34a123a221a222,a159a221,a210a52a134a34a123a221
a222,a89a199a63a135a243a230a212a166a176,a163a83a243a26a166a209a139a97a126,a139a30a127,a28a252a14a222a123a192a220,
a7a175a199a139a245a76,a15a88a21a178a124a35a13,a52a88a80a222a123,a149a52a210a52a126a23a35a98a89a199a192a220
a123a34a42a143a221a222,a117a222a89a199a252a14a222a123a236a24a92a73a123a76a66,a89a24a92a73a106a166a24a196a119a41
a232a73a1a91,a88a52a135a73a1a91,a232a73a1a91a210a52a207a196a89a74a232a73a123a7a248a166a117a159,a232
a73a1a91a243a26a166a222a139a105a76a123a97a126,a139a105a245a229a236a102a126a102a126a89a174a88,a79a143a28a7a23
a124a123a62a14,a15a88a21a252a89a199a62a14a117a222,a210a252a34a243a199,a149a243a199a123a72a42a117a23a62a14a123
7
a76a66a249a217a199a123a117a159,a89a210a52a232a73a1a91,a0a143a183a25a243a128,(3.12)a52Croftona218a42,
a210a52a128a69a193a222a30a15a139a1a91,a52a69a193a1a91,a199a123a154a183a209a127a119,a199a123a134a34a210a52a76
a203,a196a49a25a243a183a243a69a193a222a139a30a77a34,a203a196a139a76a203a209a35a98,a0a143,a89a171a134a34a123a199
a106a139a24a199a222a221,Croftona218a42a89,a199a52a127a1494a245a199a123a222a221,Feuchlla218a42a52a69a193
a222Croftona218a42a123a24a199a136a42,a79a143Croftona218a42a128a199a123a222a221a524a245a149a199a123a143a30
a52a159,a196a49a210a188a126a89a30a77a34a203a154a24a199a76a203a209a35a98a123a117a159a210a122a116Feuchlla218a42.
a79a36a175a11a210a52a128a31a143a52a69a193a222a76a203a123a222a221,a69a193a222a123a76a203a52a69a193a222a58a78a1
a91a123a134a34,a13a233a149a134a34,a183a15a139a199a222a221(measure).a243a69a193a222a123a222a221a105a59a92,a89
a52a79a143a69a193a222a123a134a34a203a244a154a139a199a59a92a123a233a83a7a248,a79a143a28a139a24a199a76a203,a199a139
a24a199a244a154,a79a36a89a30a134a34a203a89a199a244a154a139a199a233a83a7a248,a149a52a89a199a76a203a123a222a221a210
a82a143a89a199a244a154a123a222a221,a196a49a183a25a243a76a150a89a24a89a14a143a248a121a210a69a193a222a123Crofton
a218a42,a89a199a121a210a217a34a105a59a92,a88a44a7a66a101a24a154,a210a52a46a27a6a166a166a43a41a210a113a234.
a25a243a7a48a128,γa89a30a77a34a203a24a199a76a203a35a98,a0a143a89a199a76a203a44a49a166a43a41,a152a42a166
a43a41,a132a149e1(s)a89a30a77a34a203a89a199a76a203a35a98,a13a76a203a139a199a244a154y,a196a49a7a166ya123
a43a41,a0a143a166a43a41a166a31a143a17a218a89a199a76a203a166a134a34a35a98,a13a76a203a522a145a123a56a45,a196
a49a7a139a220a199a43a41,a183a210a82a89a30a77a34a123a140a159sa42a143a24a199a43a41,a0a143,a89a199a76a203a203
a134a34a35a98a123a155,a210a139a67a222a123a89a199a60a89a213a164a98a24a199a43a41a52a31a143a17a218a76a203a139a24
a199a244a154,a89a199a244a154a52y,a23a108ya52a166e1a18a134a123,a196a49y a24a189a243e2,e3 a196a196a123a203a177
a222,a89a52a79a143e2,e3a52a203e1 a18a134a123,a79a36ya210a52a243e2,e3a123a203a177a222,a149a52ya210a44a49
a44a143
y = cosθe2(s) + sinθe3(s),(3.13)
a34a19a222,ya52e2,e3 a123a24a199a34a117a28a92,a145a79a143a199a52a199a92a160a37a222,a196a49a44a49a85a196
a89a248a123a111a231,a79a36ya89a154a139a220a199a43a41,a183a44a49a82a143s,θ,a0a143a25a243a123a175a11a52a70y
a243a89a199a193a232a195a163(element area)a123a221a222,a52a7a178y a89a154a123a221a222a23a196a52a76a203a123
a221a222,a47a158a89a88a10a70,y a52a24a199a154,a28a210a70dy,ya52a243a92a160a69a222a123a24a154,a149a52a5
a193a117a70dy,a178a89a199dya85a143a203y a18a134a123a220a199a48a53a123a60a106,a0a143a89a220a199a92a160a37a222
a248a106a123a105a232a210a52y a123a221a222,a196a49a183a25a243a89a143a41,e1,e2,e3a52a41a37,a199a162a174a199a37
8
a222a52a144a35a18a134a123a92a160a37a222,a199a162a15a209a52sa123a60a106,a196a49a183a178a199a123a218a42a85a241a117:
de1
ds = a2e2 + a3e3
de2
ds =?a2e1 + a1e3 (3.14)
de3
ds =?a3e1?a1e2
a195a88a152a91,a89a218a42a176a62a123a248a106a196a24a199a39a233a193a123a48a106,a89a210a41a5a86,a105a59a92a143,
a152a42a174a24a174dy,a210a122a116dya24a199a218a42:
dy = (?sinθe2 + cosθe3)(dθ + a3ds)?e1(a1 cosθ + a3 sinθ)ds,(3.15)
a149a52a183a162a139
a193a232a195a163= (a2 cosθ + a3 sinθ)dθ ∧ds,(3.16)
a165a215
a2 = cosτ,a3 = sinτ,(3.17)
a11
a193a232a195a163= cos(τ?θ)dθ ∧ds,(3.18)
a253a33a128cos(τ? θ)dθ ∧ ds a52a69a193a222a123a193a232a195a163,a143a234a7a70a89a199a69a193a222a123a193
a232,a70a89a199a192a220a123a173a232a73,a15a210a52a70a89a199a42a6a123a173a232a73,a0a143a183a106a89a199a154a123
a30a127,a52a31a68a113a92a253a233a138,a117a70a89a199a253a233a138a123a89a199a173a232a73,a183a166a98a34a247a139a220
a199a35a106s a203θ,a0a143a210a233a199a162a117a70a232a73,a232a73a123a30a127a11a251a189s,a82cosa123a253a233a138
a117a70a232a73,a34a38a110a221a221a24a87a123a155,|cos|a24a225a35a234a245a232a17a218cosa123a253a233a138a123a232
a73a524,a89a52a79a143a104cosa440a116pia30a52a441a116-1,a24a225a524a391,a196a49a524,a79a36a89
a248a28a210a70a116a203a77a34a35a98a123a76a203a123a221a222,a253a139a245a232a199a76a203,a89a199a221a222a52a70a89a199
a173a232a73,a178a199a174a241a117,a174a241a117a131a128,a28a144a7a70a24a39a233θa123a232a73a210a44a49a234,a16a5
a123a52a77a34a123a140a159,a196a49Cr oftona175a11a210a21a44a89a199a15a174a122a116,a28a162a168a5a117,a195
a86a46a24a46,a210a44a49a122a116,a79a36Croftona218a42a128γa89a30a77a34a150a232a139a245a159,a199a178a89
a199a77a34a203a127a91a76a203a35a98a123a117a159a44a143a24a199a221a222a123a117a159,a253a139a245a159,a132a89a199a183
a210a122a116Feuchlla189a174,a89a52a105a164a224a123a24a199a189a174,a183a162a34a38a89a199a22a111a52a85a20a123,
9
a2432a145a123a60a111,a210a52a85a20a123a77a34,a233a149a85a20a123a77a34,a199a123a15a77a71a139a24a199a24a189a123
a5a33,a31a143a30a127a89a199a5a33a21a72a116? a23a108a104a89a30a77a34a52a178a193a77a34a63a37a52a24a30
a113a247(complete)a77a34a30a52a44a49a72a116a123,a202a226a222a193a255a88a15a44a49a121a210a89a199a136a88.
a205a139a63a135a123a24a199a136a42a52a196a162a123Fray-Milnor a189a174a213a152a42a77a34Ca139a136,a11
integraldisplay
C
kds ≥ 4pi,(3.19)
a183a46Milnora52a163a74a35a117a155a41a33a128a132a123a24a199a95a192a166a27a86,a198a41a89a199a189a174a123a30
a127,a210a203a28a162a24a248,a222a49a30,a144a19a225a116a89a199a175a11,a198a201a241a24a199a30a71a31a143a30a127
a24a30a85a20a123a77a34a21a75a196a24a199a136,a34a38a139a136a123a155,a23a108a183a162a77a127a137a7a77a71a205a245
a24a74,a79a143a75a136a210a122a221a221,a88a44Fray -Milnora189a174a210a52a89a34a38a89a199a85a20a123a77
a34a139a24a199a136a123a155,a199a123a92a77a71a123a232a73a150a232a1434,a196a49a92a77a71a24a189a210a76a1494,a89
a199a121a210a105a59a92,a79a143a152a42a199a66a1494a123a155,a24a189a139a199a48a53,a243a89a199a48a53a222a123a176a221
a144a139a24a199a244a76a203a24a199a244a66a154,a196a49a183a89a30a77a34a24a189a52a203a122a89a199a244a76a154a166a244
a66a154a123a220a199a140,a0a143a165a45a210a150a139a244a76a154a90a244a66a154,a196a49a165a45a0a74a178a113a123a178a193
a203a77a34a35a98a123a155,a209a35a98a149a220a154,a0a143a126a24a30a34a227a178a89a220a154a203a229a117,a149a52a199
a89a77a34a210a140a196a24a199a75a173,a23a108a77a34a210a44a49a192a196a24a154,a196a49a210a88a52a136,a196a49a152
a42a89a92a77a71a66a1494 a123a155,a199a89a30a77a34a123a136a210a44a49a137a13,a79a36a152a42a77a34a139a199a136
a123a155,a199a123a92a77a71a24a189≥ 4,a149a52a193a47a122a241a117Fray-Milnor a189a174,a5a193a183a89a89
a166a189a41a98a123a211a42a123a7a248,a189a41a98a123a211a42a52a255a88a89a199a136,a159a113a88a144a24a30a77a34,
a44a49a139a80a1a30a77a34,a253a196a162linka123a207a196,a255a88a31a143a30a127a89a199link a21a119a44a117a77
a71a52a12a71,a89a199a175a11a212a174a166a27a105a161a108a85,a33a163,a41a234a105a245a89a48a193a123a211a42,a88
a44,a77a18Fary-Milnor a189a174a139a199a44a21a117,a210a52a255a88a35a34a123a77a71a203a12a71a233a149a189
a41a98a123a88a35a42a139a31a143a107a47,a189a41a98a52a207a196a77a34a123a233a83a51a174a123a117a159a203a135a73a42
a123a7a248,a196a49a93a116a126a243a89a199a48a193,a28a88a68a122a21a178a89a199a136a137a13,a98a52a139a44a21a104
a77a34a123a77a71a166a12a71a139a236a171a117a159a123a30a127,a198a196a189a123a89a199a88a35a42a204a127a1490,a196a49
a139a74a175a11a207a196a123a30a127a44a49a46a24a46,a152a42a139a124a38a234Harrisa115a71a123a49a123a155,a198a210
a243a89a189a41a98a123a211a42,a189a41a98a52a163a1a35a117a165a41a151a9a123a24a199a58a158a128a132a123a106a166a27.
a198a150a139a214,a88a209a166a152a144a139a31a143a7a248,a24a199a124a41a198a123a11a173,a41a58a158a1a79,a58a158a13
a99a123a211a42.
10
a145a143a89 a77a193a88(a24)
2001a3511a2192a134
1 a77a193a123a41a37
a152a21a183a162a89a24a154a77a193a88,a135a232a73a243a77a193a222a97a126a123a207a196a243a114a199a106a166a176a62a52a105
a7a155a123,a89a52a79a143a243a77a193a88a165,a77a193a123a89a74a117a159a128a128a106a223a116a217a198a205a18a123a60
a111,a13a89a74a205a18a123a60a111a35a154a116a77a193a123a30a127a15a139a105a245a117a159,a243a77a193a123a60a111a46a178
a21a9a234,a0a143,a77a193a139a199a128a154,a210a52a183a162a34a189a199a52a2433a145a56a45a176a62,a196a49a28a23
a122a68,a28a44a49a152a67,a44a49a243a23a122a68a199a222a62a123a77a34a176a123a117a159a249a217a198a31a143a123,a24
a116a176a145a49a128,a210a23a88a68a234,a183a123a89a27a203a86a88a24a248,a196a49a183a46a76a27a178a89a1a16a97
a238a133a92a24a5,a89a199a97a238a76a150a97a148a243a202a47a135a73a1a91a86a222a73a88a116a123,a199a139a199a128
a154,a210a52a79a122a105a13a37a48a27a117a122a59a92,a0a143a31a143a52a77a193a17? a77a193a210a52a67a222a24
a199a59a77a123a192a220,a183a178a199a123a154a123a43a41a44a143a220a199a35a106a123a60a106,a89a220a199a35a106a183a119
a41u,v,u,va24a196a119a41a107a106(parameter),a34a38u,va35a154a123a30a127,a89a74a154a123a27a236
a210a196a234a24a1992a145a123a77a193x(u,v) = (x1(u,v),x2(u,v),x3(u,v)),a149a52a79a143a139u,v,
a196a49a28a44a49a38a122a24a199a107a106a123a138a52a158a106,a108a128a38a122a12a24a199a107a106a35a154,a247v a52
a158a106,a13u a35a154,a196a49a210a139a24a30a77a34,a199a123a107a106a52u,a51a248,a28a44a49a38a122ua88
a35,a13va35a154,a79a36a243a77a193a176a62,a139a220a28a77a34,a199a123a107a106a24a28a52v a127a149a158a106,a24
a28ua52a127a149a158a106,a233a149a89a220a28a77a34,a154a24a30a77a34a243a154a24a154a209a139a24a30a35a34,a196a49
a243a24a199a154x,a183a162a210a139a220a30a134a34,a183a34a189a89a220a30a134a34a88a173a92,a166a233a155a128,a137a219
a143a89,xu,xv a88a52a51a24a199a48a53,a88a178a134(a34a117)a35a7,a217a165xu,xva210a52xa123a37a222a73
a47a233u,va70a160a135a73,a221a86a183a128,a199a123a37a222a232xu ×xv negationslash= 0,a34a38a89a220a199a48a53a88a173
a92,a196a49a199a162a210a60a196a24a199a178a193,a89a199a178a193a183a119a41a77a193a243a89a24a154a123a35a193,a89a199
a35a193a243xa89a24a154a139a199a18a134a123a48a53,a89a199a48a53a123a134a34a24a196a119a41a27a34,a215a248a27a34
a123a48a53a139a24a199a92a160a37a222,a79a36a15a119a41a27a37a222,a89a199a27a37a222a139a220a199a160a10,a199a44
a49a53a222a18,a15a44a49a53a5a18,a139a220a199a48a53a166a80a35a39a123a160a10,a183a160a10a199a38a122xu,xv
a203a27a37a222a52a24a199a141a67a123a43a41a41a37,a52a24a199a141a67a248,a166a233a155a128,a183a119a89a199a27a37
1
a222e3,a63a34a247e3a52a199a92a160a37a222,a149a52e3a210a119a22a30a71
(e3,e3) = 1 (a92a160a37a222),e3 = xu ×xv|x
u ×xv|2
,(4.1)
a0a143a243a89a248a123a160a10a131a5,(xu,xv,e3)a210a52a141a67a248,a89a30,a113a239a42det(xu,xv,e3)a52
a116a123,a253det(xu,xv,e3) > 0,a196a49a52a141a67a248,a25a243,a183a123a89a199a48a27a203a24a196a123a48
a27a52a88a51a123,a24a196a123a86a222a128a128a126u,v a107a106a21a48a114a199a123a77a193a123a135a73a1a91,a79a36
a210a7a118a159a234,a198a162a89a176a139a24a199a98a154,a79a143xu a203xv a88a24a189a18a134,a0a143a183a162a123a108
a85a52a243a149Euclid a1a91a139a24a199a221a222,a196a49a198a162a126a123a52a58a18a134a123a43a41a248,a13a1a91
a52a24a248a123,a98a52a73a219a48a193a123a218a42a210a7a118a133a236a234,a13a183a82e1,e2a52a92a160a37a222,a199
a162a52a144a35a18a134a123,a196a49a25a243e1,e2,e3a174a199a37a222a209a52a92a160a37a222,a13a37a144a35a18a134,
a63a37a79a143a7a199a52a24a199a141a67a248,a196a49a199a123a113a239a42a97a148a52a116a123,a98a52a79a143a89a174a199
a37a222a209a52a144a35a18a134a123a92a160a37a222,a196a49a113a239a42a127a149±1,a199a123a178a48a127a1491,a196a49
a183a25a243a52a119a89a199a113a239a42a127a1491 a123,a79a36a89a52a24a199a116a98a123a43a41a248,a199a123a113a239a42
a127a1491,a149a52
e1,e2,e3 (a141a116a98a41a37),(ei,ej) = δij,(e1,e2,e3) = 1,1 ≤ i,j,k ≤ 3,(4.2)
2 a77a193a123a135a73a42a249a217a1a91
a0a143a135a73a1a91a14a143a248a17? a89a30a210a88a144a52a139a24a199a43a41a248,a13a52a139a24a24(family)a43
a41a248,a164a139a1a199a35a106(a35a123a107a106),a0a143a33a7a155a123a24a199a25a54a210a52a24a199a43a41a248a203
a199a248a163a123a43a41a248a52a14a143a24a199a7a248,a7a234a137a89a199a7a248a52a135a73a1a91a33a204a7a123a175
a11,a196a49a183a25a243a139a24a24a43a41a248,a7a48a128a52a139a220a199a35a106u,v,a4a150a44a49a139a245a199a35
a106,a0a143a7a73a89a199a248a163a43a41a248a203a199a123a7a248,a183a210a178a199a135a73a234,a25a243a183a89a199xa52
a37a222,eia15a209a52a37a222,a196a49a183a210a70a70a23dx,a23da203dei,a89a52a24a199a37a222a123a135a73.a98
a52a79a143e1,e2,e3 a52a24a199a41a37,a52a34a117a195a7a123,a13a183a162a52a2433a145a56a45a165a255a88a123,
a196a49a127a91a24a199a37a222a23a108a52e1,e2,e3 a123a34a117a28a92,a196a49a183a44a49a178deia85a196
dei = ωijej,(4.3)
2
a89a176a183a126a123a52Einsteina123a110a82,a152a42a139a24a199a141a106a173a133a123a155a210a35a28,a79a143a183a162
a123a56a45a523 a145a123,i,j,ka209a52a441a1163,a0a143ωijej a210a52a233j a35a28,a196a49a89a523 a49,
a253j = 1,2,3,a1393a49,a89a52Einstein a243a135a73a1a91a90a159a123a110a82,Einsteina164a41a234
a24a71a47a60,a7a48a128,a44a3a28a7a139a199a106a248,a199a7a139a199a141a106,a253xi,ia89a199a141a106a209a85
a196a5a41,Einsteina128a88a85a5a41,a198a210a85a234a222a41,a79a36a128a117a123a135a73a1a91a123a86a176
a62,a222a41a58a158a245,a43a41a123a89a199a141a41a209a85a196a222a41,a217a34,a222a5a150a139a7a248,a196a49,
a183a126a5a41a85a196a218a42(4.3),a218a42(4.3)a52a228a253a123a218a42,a199a44a43a220a199a248a163a43a41a123
a7a248,a24a199dei a203a198a117a123a43a41ei a123a7a248,ωija52a31a143a17? a199a52a24a39a135a73a42,a79
a143dei a52a24a39a135a73a42,a199a123a138a52a37a222a123a24a39a135a73a42,a196a49a28a152a42a85a196a199a123a73
a222a123a155,a210a1393a199a73a222,a13a37a154a24a199a209a52deia123a73a222,a0a143ωija52a24a39a135a73a42,a79
a143i,ja209a52a441 a1163,a196a49a24a225a1399a199,a98a52a89a74ωij a88a52a127a63a123,a89a52a1a91a222,
a197a166a222a33a228a253a123a24a199a218a42,a79a143a183a162a24a35a209a52a116a98a123a248a58,a196a49ωija233a149a5
a41i,ja52a39a233a193a123,a253
ωij + ωji = 0,(4.4)
a143a31a143a17? a79a143ei,ej a119a22(ei,ej) = δij,a28a178a199a135a73,a145a79a143 deltaij a52a158a106,
a196a49a135a73a199a131a128a520a199a149a52a210a21a122a116a5a193a218a42:
(dei,ej) + (ei,dej) = 0,(4.5)
a132a149dei = ωikek,a63a37ei,eka52a144a35a116a98a123,a196a49a132a222a42a210a122a116ωij+ωji = 0,a253
a218a42(4.4),a233a149a196a139a123i,ja441a1163,ωija52a39a233a193a123,a79a36(ωij)a23a196a24a1993×3a123
a48a106,a0a143a89a199a48a106a52a39a233a193a123,a199a88a52a1399a199a195a163,a34a19a222,a144a1393a199,a63a37
a79a143a39a233a193,a196a49a204(a233a110)a34a222a123a195a163a209a520,a217a198a123a233a149a204a34a52a39a233a193a123.
a15a210a52a139
(ωij) =
0 ω12 ω13
ω12 0 ω23
ω13?ω23 0
,(4.6)
a79a36a144a16a53a199a195a163,a253a144a139ω12,ω13,ω23,a893a199a209a52a24a39a135a73a42,a13a893a199
a24a39a135a73a42a233a77a193a1a91a117a159a52a58a158a7a155a123,a253a199a162a209a44a49a132a135a73a42a117a44a43.
a202a47a207a196a60a106a88,a180a73a219a123a30a127,a209a52a89a60a106,a255a88a60a106a221a86a52a178a60a106a135a73
3
a234,a178a199a123a135a219a42a143a248a106,a76a27a4a58a88a243a14a149a24a39a135a73a42,a217a34,a24a39a135a73
a42a52a178a89a199a175a11a65a122a59a92a234,a183a13a62a123a30a127a17a178a203a28a162a144a235a44a27a41a123a76a106
a166a27Darbouxa123a86,a199a139a143a76a253a119a41a21a77a193a88a22,a89a171a86a105a138a122a23,a88a44a44
a234a52a27a169a123,a198a88a126a135a73a42,a198a126a123a52a160a135a73,a196a49a139a142a245a218a42a85a229a117a159a24
a74,a126a135a73a42a123a155,a24a39a135a73a42a85a229a117a210a59a92a245a234,a196a49a183a89a176a126a234a24a39a135
a73a42,a126a116a98a41a37,a38a77a193a88a58a158a59a92,a89a74a28a162a243a202a47a135a73a1a91a86a165a105a232
a21a73a116,a98a52a89a171a48a27a105a139a72,a79a143a24a35a192a220a209a59a92a234,a183a34a189a77a193a52a189a53
a123,a253a243a221a123a30a127a139a24a199a39a30a168a48a53,a79a143a189a234a53a131a128,a243a24a199a154,a199a139a220a199
a35a37a222a123a155,a199a123a27a37a222a210a113a92a104a189a234,a79a36,a77a193a189a53a131a128,a154a24a154a24a189a139
a24a199a251a189a123a92a160a27a37a222,a88a52a199a123a139a123a37a222,a0a143,a77a193a222a139a24a199a105a7a155a123
a1a91a136a232,a210a52a24a199a154a28a24a199a92a160a35a37a222,a253xa203e1,a89a52a25a83a196a162a14a145a45a123
a33a59a92a123a60a89,a15a52a33a7a155a123a60a89,a0a143a233a149x + e1,a199a245a234a24a199a145a106,a79a143
a251a189a234xa131a128,e1a89a199a92a160a35a37a222a164a44a49a221a87,a196a49xa123a27a236a522a145a123,a0a143a154
a24a199xa123a35a37a222a164a122a281a145,a89a52a79a143a199a44a49a52a24a199a203,a221a24a199a203a177,a199a52a92
a160a123,a196a49a89a199a56a45a523a145a123,a13a89a1993a145a56a45a183a119a41E:
E = {xe1|x ∈ M}(a203a45); dimE = 3,(4.7)
a139a24a1993a145a123a56a45a221a86a128a5a241a24a1993a145a56a45,a89a199a10a39a7a155a244a234,a25a243a142a245a106
a166,a212a174a209a137a7a89a199a10a39,a24a96a28a126a198a117a123a22a111a117a195a85a1a91a88a234,a128a128a137a7
a222a193a139a24a199a203a87,a89a199a183a162a119a41a14a145a45,a89a30a203a177a52a24a199a14a145,a79a36a89a199a14
a145a45a119a41a203a45,a79a143a14a145a52a203,a196a49Ea52a132a22a111M(a77a193a183a119a199a143M,a52a24a199
a22a111)a5a241a117a123a24a199a203a45,a0a143a243a24a154a7a139e1,a210a139e2a234,a89a52a79a143e2 a52a203a199
a18a134a123a24a199a92a160a35a37a222,a51a30a44a49a132a198a117a123a189a53a104a189a5a117,a79a36e1,e2a210a189
a5a117a234,e3a52a27a37a222a104a108a15a189a234,a196a49xe1a89a199a92a160a35a37a222a203a41a37a52a24a195a47.
a139a234a92a160a35a37a222a15a44a49a232a5a24a199a41a37,a104a108a139a234a41a37,a28a210a44a49a82a145a24a199a35
a37a222a143e1,a196a49a893a145a56a45a210a52a183a162a123a77a193a196a139a89a74a41a37e1,e2,e3 a196a196a123a56
a45,a149a52a183a162a210a139a222a193a123a218a42(4.7),a183a128Ea523a145a123,Ea139a24a199a110a243,a110a116a198
a117a123a77a193M,a79a143a28a139a89a199a35a37a222,a199a139a24a199a198a154x,a132xe1a178a199a110a143x,a89a52
a24a199a44E a116Ma123a110a243,a7a207a196a77a193a123a135a73a1a91,a92a44a77a193a88a234,a24a189a7a126E.
4
Ea203a198a117a123a77a193a139a183a35a123a7a248,a166a98a183a89a234a89a171a1a91a123a7a248,a126Ea123a80a255a243
a149,E a123a56a45a523a145a123a56a45,a199a222a62a139a24a39a135a73a42,a89a74a135a73a42a243Ea222a62a209a52
a104a189a123,a0a143a248a234a183a89a123e1,e2,e3a89a1a199a92a160a37a222,a104a108a77a193a123a154xa15a52a24a199
a37a222,a199a123a160a152a37a222a15a52a24a199a37a222,x a52u,va123a60a106,dx a104a108a52u,v a123a24a199
a24a39a135a73a42,a199a15a44a49a44a143e1,e2,e3 a123a24a199a34a117a28a92,a98a52a34a19a222,a199a24a189
a52e1,e2 a123a28a92,a89a52a79a143e3 a52a27a37a222,a196a49a199a24a189a52a243a35a193a222a62,a13e1,e2a209
a52a35a37a222,a196a49dxa24a189a52e1,e2a123a34a117a28a92,a199a123a248a106a183a119a41ω1,ω2,a253a122a116a183
a162a145a24a199a218a42:
dx = ω1e1 + ω2e2,(4.8)
a196a49a183a25a243a1395a199a24a39a135a73a42,ω1,ω2,ω13,ω23,ω12,a89a28a135a73a42a58a158a7a155,a199
a162a209a139a59a92a123a1a91a63a66,a89a199a210a128a210a135a73a42a255a88a1a91a117a159a38a122a175a11a59a92a13
a37a142a52,a25a243a183a162a210a139a218a42( 4.8),a183a162a82a218a42(4.3),(4.4)a42a143a183a162a123a145a19a199
a218a42,a183a243a3a193a89a44,a135a73a42a139a199a33a76a123a128a154,a210a52a135a73a220a39a128a520,a253a233
a149dx,d(dx) = 0,a233a149a127a91a60a106a123a105a135a73a220a39a24a189a127a1490,a89a210a35a104a149a243a56
a45a127a63a82a24a199a75a173,a242a82a199a123a30a142,a13a30a142a88a242a139a30a142,a82a30a142a82a220a39a24a189
a52a127a1490,a79a143a89a199a117a159,a89a171a106a166a136a232a210a139a196a162a123a51a174(homology)a117a159,a25
a243a28a7a180a31a143a192a220,a209a52a51a174a117a159,a58a158a7a155,a88a44a183a162a25a243a178a218a42(4.8)a38
a30dx a242da24a5a6a210a127a1490,a13a178a141a30a123a48a13a210a122d( omega1e1) + d(ω2e2),a253
d(ω1e1) + d(ω2e2) = 0,(4.9)
a213a63a104a105a135a73a3a193a139a24a199a24a39a123a155,a135a73a145a19a199a79a6a7a149a82,a15a13a211a131,a28
a210a178a199a162a135a73a234,a135a73a131a128,a210a21a25a196a122a116a123a42a6a52e1,e2,e3a123a34a117a28a92,a0
a143a199a123a248a106a52a19a39a135a73a42,a13a233a149a89a199a248a106a52a19a39a135a73a42a123a37a222a7a127a1490a123
a155,a196a139a123a248a106a209a7a127a1490,a149a52a28a152a42a13e1,e2a123a248a106a1430 a123a155,a210a122a116a183a5
a193a123a145a174a199a218a42:
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(4.10)
a183a88a45a251a178a199a41a234,a89a199a121a210a105a59a92,a44a193a253a44a122,a89a52a7a155a244a234a123a24a199a218
a42,a89a74a218a42a28a162a15a142a250a122a99,a79a143a243a202a47a86a222a23a88a68,a89a52a132a149a202a47a88
5
a245a161a126a135a73a42,a142a245a124a15a88a204a126a135a73a42,a217a34a89a105a59a92,a196a49a183a210a122a116a218
a42(4.10),a108a128a13e3a123a248a106a1430,a210a122a116a145a143a199a218a42:
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(4.11)
a89a52a58a158a58a158a7a155a123a218a42,a132a89a74a210a122a116a196a162a123Levi-Civitaa178a113a117,a25a243,
a89a199a15a119a201a100(connection),a233a149a201a100,a202a47a73a24a253a86,a44a49a89a222a105a200a105a200,
a217a34a105a59a92,a183a128a243a89a199a60a111a131a5,Levi-C ivitaa201a100a210a52ω12a89a199a24a39a135a73
a42,a213a63ω12 a52Ea176a62a123a24a199a24a39a135a73a42,a199a210a189a1a91a123a117a159,a38a122a183a44a49a178
a89a199a37a222a215a248a24a30a77a34a178a113a123a35a196,a89a52a31a143a63a135a17? a210a52ω12a89a199a24a39a135
a73a42a132a48a199(4.10)a113a92a104a189,a79a143a152a42a139a24a199ωprime12,a38a122a55a92a51a248a123a48a199a42,
a253
dω1 =?ω2 ∧ωprime12; dω2 = ω1 ∧ωprime12,(4.12)
a183a137a7a121a210ω12 = ωprime12,a253a144a139a24a199a44a21a117,a144a139a24a199ω12 a55a92a48a199(4.10),a15
a210a52a128,a34a38a139a12a105a24a199ωprime12a55a92a48a199(4.10),a0a143a183a162a178a220a199a48a199a35a62,a210
a122a116
ω1 ∧(ωprime12?ω12) = 0; ω2 ∧(ωprime12?ω12) = 0,(4.13)
a13a220a199a24a39a135a73a42a152a42a35a198a127a1490a123a155,a13a37a152a42a217a165a24a199a88a520,a0a143a217
a199a0a199a23a108a52a199a123a245a106,a89a52a105a83a106a33a59a92a123a192a220,a28a162a174a24a174a210a241a117
a234,ω1,ω2a52a88a1430 a123,a13a37a88a98a88a127a1490,a63a37a34a117a195a7,a79a36,ωprime12? ω12a17
a127a149ω1a123a245a106,a145a127a149ω2a123a245a106,a98a52ω1,ω2 a34a117a195a7,a196a49a199a7a127a1490.
a79a36a152a42a242a139a24a199ωprime12a55a92a51a248a123a48a199,a199a24a189a127a149?mega12,a89a52a113a92
a104a189a123,a0a143a183a126a89a199a90a159a196a162a123Levi-Civitaa178a113a117,a183a162a25a243a139de1 =
ω12e2 +?mega13e3,a183a162a178e3a89a24a49a82a62,a82a62a52a31a143a63a135a17? a34a38a139a24a199
a37a222,a28a178a199a123e3a82a62a131a128,a210a52a215a248e3a123a48a53a82a89a199a116a98a61a107,a196a49a28a178a199
a82a62a234a123a63a135a210a52a82a199a123a116a98a61a107,a28a178a199a82a62a123a155,a183a25a243a88a242a52a202a47a123
a135a73a234,a52a99a123a135a73a234,a183a126Da44a43a89a199a99a123a135a73,a149a52a210a122a116a89a28a218a42:
De1 = ω12e2,De2 =?ω12e1,(4.14)
6
a89a52a76a183a166a98a121a210a123a52a132a1a91a113a92a104a189a123,a89a52a79a143ω12a52a113a92a104a189a123,a34
a38De1 = 0a123a155,a183a210a128a89a199a37a222a52Levi-Civitaa63a66a5a178a113,a253a183a128a243a89a171a60
a111a5a199a210a52a178a113a123,a199a52a178a113a123,a89a52a139a99a123a63a66,a243Euclid a1a91a123a30a127,a89
a210a52a202a47a178a113,a25a243a52a202a47a178a113a123a77a18,a15a210a52De1 = 0a30a52Levi-Civitaa178
a113,a196a49a243a77a193a222,a28a126a24a39a135a73a42,a166a11a89a199a178a113a7a118a10a195,a13De1 = 0a123
a155,a233a149a37a222a52a199a135a73a48a199,a89a199a135a73a48a199a88a24a189a139a137,a98a52a34a38a28a139a24a30
a77a34,a28a210a44a49a215a248a89a30a77a34a70a137,a233a149a77a34a117a128,a196a139a123a89a74a60a106a209a52ta123
a60a106,a215a248a89a30a77a34a70a137,a44a49a70a116a137,a150a149a89a199a30a71,a183a128a89a199a37a222a210
a52Levi-Civita a178a113,a196a49a44a89a199a189a66a44a49a234a137a89a199a178a113a117a203a77a34a123a160a10a139
a7,a34a38a139a24a199a154,a12a105a164a139a24a154,a0a143a28a201a89a220a154a139a220a30a88a51a123a77a34,a28a178
a51a24a199a37a222a215a62a24a30a77a34a178a113,a63a215a145a19a30a77a34a178a113,a23a116a123a24a196a88a52a51a24
a199a37a222,a150a149a89a199a178a113a117a166a77a34a123a160a10a139a7a248,a52a202a47a123a178a113a117a123a77a18,a202
a47a123a30a127,a178a113a117a243Euclid a178a193a176a222a52a253a233a123a90a189a234a123,a25a243a89a199a30a127a52a203
a77a34a123a160a10a139a7a248,a89a210a52a201a100,a183a166a98a178x a89a199a37a222da220a39a127a1490,a13a51a248
a123,a183a44a49a233eia89a199a37a222a15d a220a39,a253d(dei) = 0,a79a36a174a241a117d(dei) = 0,a210
a139:
dωij = ωik ∧ωkj,(4.15)
a34a19a222,a105a59a92a143,a183a233dei = ωijej a70a135a73,a108a128a79a143ωija52a24a39a135a73a42,a196a49
a97a148a139a199a139a82,a253a139a123ωij ∧dej,a149a52a139
d(dei) = dωijej?ωij ∧dej = 0,(4.16)
a98a52dej = ωjkek,a196a49a62a24a199dωijeja149a143dωikek a150a139a7a248,a89a52a79a143a28a233
a149j,ka52a70a90,a44a85a196j,a15a44a85a196k,a209a52a441a1163,a195a196a162a123,a79a36a210a122a116
a218a42(4.15),a89a52a24a199a228a253a123a218a42,a23a248a102a38,a217a34a105a59a92,a131a217a52a2433a145
a123a60a111,a105a59a92,a34a19a222,a141a30a89a229a117a523a49a131a90,a98a52a893a49a52a233a149j a70
a90,a79a143ω a52a39a233a193a123,a196a49a183a162a117a23ωik,i negationslash= k,a221a86a210a117a23a23ω13,a243a5
a193a0a199a218a42a23d omega13 a97a148a52ω1jωj3,a98a52a34a19a222,ja144a21a127a1492,a89a52a79
a143j = 1a30ω11 = 0,j = 3 a30ω33 = 0,a196a49a141a30a23a248a52a89a143a248a131a90,a217a34a210a144
7
a139a24a49,a196a49a28a210a122a116
dω13 = ω12 ∧ω23,(4.17)
a51a248a44a49a122a116
dω23 =?ω12 ∧ω13,(4.18)
a89a74a24a196a210a52a196a162a123Weingartena218a42,a58a158a59a92,a196a49a89a74a218a42a209a52a44a59a92
a123a105a135a73a193a253a122a116a123,Weingartena218a42a105a139a63a135,a28a7a199a203a198a117a123ω1,ω2a123
a218a42a7a118a123a155,a113a92a52a24a199a111a231,a253a213a63a116a89a74a218a42a166a218a42(4.10)a35a145.a89a199
a111a231a52a139a1a91a63a66a123,a25a243a199a126a117a189a196a162a123Betti Transform ation,a89a199a183a88
a251a89a234,Weingartena218a42a203a198a117dωia123a218a42a123a35a145a117a52a139a253a47a123a1a91a63a66a123.
3 a77a193a123a228a253a88a35a42
a222a193a183a255a88a234De1a123Levi-Civita a178a113a117,a152a42a82a89a199a37a222a243e3a89a199a48a53a123
a155,a183a210a122a116ω1 ∧ω13 +ω2 ∧ω23 = 0,a253a218a42(4.1 1),a79a36,ω13,ω23a209a52ω1,ω2
a123a34a117a28a92,a183a44a49a85a196
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(4.19)
a132a89a199a183a122a116a89a199a77a193a123a7a155a123a88a35a42,a24a199a77a193a166a12a24a199a77a193a139a31a143a73
a47? a89a199a73a47a243a149a77a71,a77a71a52a77a193a222a123a60a106,a7a48a128,a24a196a123a155,a183a162a126a145
a24a228a253a42a203a145a19a228a253a42a117a44a43:
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (4.20)
a145a19a228a253a42 II = (?dx,de3) = aω2 + 2bω1ω2 + cω22,(4.21)
a233a149a145a24a228a253a42,a183a119a199(I).a199a210a52a77a193a123ds2 = (dx,dx),a132a149dx =
ω1e1 + omega2e2,a98a52e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49a210a122a116ω21 + ω22.
a79a36ωa895a199a24a39a135a73a42a209a139a173a7a123a1a91a63a66,ω1,ω2 a123a178a48a90a210a52a77a193a123
a221a222,a2433a145a56a45a176a62,a77a193a104a108a139a24a199a221a222,a0a143a199a123a221a222a52a31a143a17? a210
a52a199a123Riemann a221a222,a52a24a1992a39a123a135a73a42,a89a1992a39a135a73a42a59a92a244a234,a210
8
a52ω21 + ω22,a0a143,a183a15a44a49a233e3a89a199a37a222a82?(de3,dx),a76a183a166a98a196a85a123,a199
a210a127a149aω2 + 2bω1ω2 + cω22,a89a15a52a24a1992a39a135a73a42,a89a199a24a196a119a41a145a19a228a253
a42,a0a143a139a24a199a145a24a228a253a42,a164a139a199a145a19a228a253a42,a28a210a82a199a123a1a110a138,a1a110
a138a123a90a210a52a + c,a1a110a138a123a232a210a52ac? b2,a24a196a143,H = a+c2 a119a41a77a193a123a165
a77a71,K = ac? b2 a119a41a77a193a123Gaussa77a71,a89a176a7a155a244a234,Gaussa77a71a139a142
a245a139a85a123a117a159,a79a36a89a52a14a143a248a445a199a24a39a135a73a42a132a199a162a123a34a117a28a92a123a7a248
a210a122a116a77a193a123a88a35a42,a165a77a71a203Gaussa77a71a89a220a199a88a35a42a195a85a77a193a123a1a91
a117a159,a7a48a128,a44a49a121a210Gauss a77a71a52a116a123a155,a77a193a52a243a123,Gaussa77a71a52a139
a123a155,a77a193a210a139a145a154,a210a54a106a145a154,a142a245a1a91a117a159a209a44a49a126a89a220a199a77a71a117a195
a85,a196a49a52a77a193a176a62a220a199a33a204a7a123a88a35a42,a89a199a88a35a42a52a60a106,a49a128a183a162
a122a116a123a52a24a39a135a73a42,a13a24a39a135a73a42a88a76a142a52a46a54a196a190a52a31a143a63a135,a98a132
a199a123a228a174a44a49a122a241a60a106a117,a89a60a106a104a108a52a234a137a122a7a118a56a249,a34a122a116a165a77a71
a203Gauss a77a71,a165a77a71a127a1490,a24a196a119a41a244a66a77a193,a196a162a123a244a66a77a193,a222a190a210
a52a28a178a24a30a85a20a123a77a34a56a243a61a6a121a176a62a196a196a123a77a193a210a52a193a232a33a66a123a77a193,
a199a123a165a77a71a1430,a196a49a89a139a59a92a123a1a91a63a66,a33a163,a4a150a25a243a209a139a105a245a7a149a244
a66a77a193a123a207a196,Gaussa77a71a205a7a155,Gaussa77a71a52a127a149ω13 ∧ ω23,a243a77a193a222a15
a24a248a139Gaussa110a243,a77a193a154a154a139a24a199a92a160a27a37a222,a178a89a199a92a160a27a37a222a23a143a24
a199a204a187a1431 a123a69a193a123a154,a210a178a77a193a110a243a116a69a193a222a86a234,a154a199a154a139a24a199a92a160a27
a37a222,a28a2430a154a152a24a199a92a160a27a37a222a203a199a178a113,a199a123a224a154a210a243a92a160a69a193a222,a0a143
a233a149a196a139a123a154a209a41a89a199a232a5a123a155,a210a243a92a160a69a193a222a122a116a24a199a75a173,a243a89a199a110
a243a5,a220a199a193a232a195a163a123a7,a253a199a123a52(imag e)a123a193a232a195a163a203a198a117a193a232a195a163a123
a7a210a52Gauss a77a71,a89a248a24a5a6a210a23a241a117a234,a196a49a89a199Gaussa77a71a139a105a59a92
a123a1a91a63a66,a89a30,a77a193a123a255a88a52a24a199a77a18,a77a193a123a30a127a139a77a34,a77a34a15a139a24
a199Gaussa110a243,a0a199Gaussa110a243a52a82a92a160a35a37a222,a217a34a15a44a49a82a92a160a27a37a222.
a0a143a77a34a123a30a127,Gaussa110a243a178a35a34a110a243a116a92a160a203a222a62,a178a92a160a203a123a221a222a250
a198a117a89a199a221a222a248a210a52a77a71,a25a243a210a178a89a199a10a39a77a18a116a176a145,a77a18a1162a145,a253a77
a18a116a77a193,a196a49a183a132a89a199a77a193Gaussa110a243a116a24a199a92a160a69a193a222a62,a89a220a199a193a232
a123a7a210a52Gaussa77a71,a196a49a89a52a24a199a58a158a7a108a123a1a91a221a222,a25a243a139a199a105a7a155
a123a7a248,a222a62a139dωik = ωij ∧ ωjk,a25a243a183a178a89a199a218a42a126a116ω12,i,k = 1,2,a149
9
a52dω12 = ω13 ∧ ω32,a98a52ωa52a39a233a193a123,a196a49ω32 =?ω23,a89a176ω13 ∧ ω23 a210
a52ac?b2,a15a210a52Gaussa77a71,a196a49a183a210a122a116a218a42
dω12 =?Kω1 ∧ω2,(4.22)
a217a165,Ka52Gaussa77a71,a89a52a24a199a13Gauss a58a158a175a199a123a218a42,a196a49Gaussa178a89
a199a189a174a119a41Theorem Egregious,Egregiousa52a110a182a9a241,a63a135a141a89a52a24a199a234
a88a122a123a189a174,a143a31a143a17? a199a121a210a234Gaussa77a71a144a203a77a193a123Riemann a221a222a139
a7,a203a89a199a77a193a243a56a45a123a160a152a195a7,a79a143a243a89a199a218a42a176a62,ω12a46a178a121a210
a234a144a203a77a193a123Riemanna221a222a139a7,a108a128ω1 ∧ ω2a210a52a243a89a199a221a222a5a123a193a232a195
a163,a104a108a144a203ds2a139a7,a253a144a203Riemanna221a222a139a7,a196a49a175a108Gaussa77a71a52a24
a199a77a193a243a56a45a176a62a123a24a199a88a35a42,a98a52a199a144a203a77a193a123Riemanna221a222a139a7.
a166a233a155a128,a28a178a77a193a35a166(deform)a24a5a6,a38a122Riemanna221a222a88a35,Gaussa77
a71a210a88a35,a196a49Gaussa77a71a139a89a143a173a7a123a117a159,a183a46Gau ssa41a234a142a245a7a155
a123a90a164a224a123a136a42,a89a199a189a174a23a108a52a198a1a47a97a220a123a24a199a136a42,a52a105a1a47a123
a24a199a136a42,a34a19a222,a183a162a210a178a89a74a48a199a126a105a135a73a135a73a24a5a6,a122a116a228a253a218
a42,a108a128a210a122a116a89a74a218a42,a5a39,a183a7a89a132a89a199a218a42a121a210Gauss-Bonneta218
a42,Gauss-Bonneta218a42a52a77a18a174a110a111a174a110a131a90a127a149180o,a82a89a199a218a42a77a18a116
a77a193a123a60a111,Gauss-Bonneta218a42a123a97a126a58a158a131a18,a121a210a123a30a127,a183a41a170a188
a126a203a45,a89a203a45a227a135a126a122a133a236a24a74,a34a189a89a199a45a52a24a199a133a123a34a45(complex
line),a0a143a89a199a133a34a45a123a14a145a52a30a133a34,a243a133a34a176a62,a253a233a138a127a1491a123a133a106
a52a24a199a203a177,a210a52a203a45,a89a199a10a39a243a212a174a222a52a228a253a123,a25a243a76a27a180a122a105a245
a123a52a98a1a91( symplectic geometry),a92a139a98a136a232(symplectic structure)a88a212
a139a126,a28a7a178a98a1a91a126a116a222a6a197a166a123a155,a137a7a28a222a24a199a133a34a45,a210a52a183a162
a25a243a196a89a123a192a220a123a24a199a77a18,a183a162a25a243a892a145,a24a116a212a174a123a155,a56a45a166a30a45
a28a243a24a229a524a145,a196a49a28a228a253a123a56a45a524a145,a2434 a145a56a45a104a165a139a24a199a133a34a45.
a79a36a210a122a116a89a199Gauss a77a71,a25a243a89a199a77a71a522a39a135a73a42,a199da24a5a6a127a1490,
a210a122a116Maxwella48a199,a196a49a135a232a73a243a1a91a222a97a126a52a142a245a212a174a123a228a250,a33a163
a183a243a21a41a166a22a236a147a222a85a234a24a159a169a57,a119a41a21Gauss-Bonneta218a42a166Maxwella48
a199a22,a28a162a152a42a21a73a116,a44a49a86a23a24a23,a183a46a178a89a159a169a57a252a117a201a76a27,a98a52
10
a25a243a79a122a105,a150a73a116,a5a39a183a46a89Gauss-Bonneta218a42a123a24a199a121a210,a89a199a203a183
a162a123a211a42a139a58a158a183a35a123a7a248,a2435 0a35a3,a243a102a210a220a8a201a76,a183a115a135a73a1a91,
a210a122a116Gauss-Bonneta218a42a123a89a199a121a210,a202a47a86a222a133a236a245a234,Gauss-Bonneta218
a42a204a7a63a135a52a128,a28a139a127a91a24a199a77a193,a77a193a222a62a139a24a199a245a30a111,a178a89a199a245
a30a111a123Eulera1a110a106a44a43a171a171a77a71a123a28a92,a89a52Gauss-Bonneta218a42,a183a122a116
a89a199a121a210,a183a46a89a52Gauss-Bonneta218a42a33a59a92a90a33a7a108a123a24a199a121a210,a210a52
a178dω12a24a135a73a210a122a116Gauss-Bonneta218a42a232a73a123a42a6,a108a128a183a243a24a202a143a1a35
a123a30a127,a116a234Princeton a41a176a145a123,a104a108a7a108a13a108a143,a183a46a27a6a178a89a199a77a18
a116a176a145,a89a199a52a44a49a41a116a123,a49a128,a183a164a41a234a74a47a123a211a42,a196a49a44a49a128a183a89
a199Gauss-Bonneta218a42a123a121a210a243
a183a196a41a123a211a42a165a52a183a33a245a161a123a24a199,a183a46a25a243a44a49a136a101a234.
11
a145a202a89 a77a193a88(a19)
a9Gauss-Bonneta218a42
2001a3511a21923a134
1 a77a193a88a21a48a123a59a144
a105a176a108a145a166a76a27a68a193a234,a183a243a26a211a176a212a234a1a21,a28a162a44a49a23a241a117a183a164a150
a139a113a92a80,a88a44a183a250a122a183a164a52a203a76a27a89a89a89a74a192a220,a0a143,a183a152a21a7a89a123
a52Gauss-Bonneta218a42,a89a199a218a42a139a35a104a123a63a66,a15a139a35a104a123a187a36,a131a217a203a183
a199a124a123a211a42a15a139a7a248,a196a49a183a7a10a24a10a183a203a89a199a175a11a52a14a143a248a123a7a248,a183
a162a222a39a89a116a77a193a88,a77a193a88a52a135a73a1a91a176a62a33a173a7a123a24a92a73,a79a143a142a245a135
a73a1a91a123a25a54a243a2433a145a56a45a176a1232a145a77a193a123a60a89a46a178a151a9a234,a51a30,a79a143a199
a52a2433a145a56a45a176a62,a89a199a1a91a123a60a89a52a44a49a23a68a123,a88a52a113a92a126a83a106a117a44a43.
a77a193a88a139a105a159a123a187a36,a33a0a123a104a108a52Monge,Monge a52a27a41a123a76a106a166a27,a198
a144a11a9a233a117a163a139a74a217a196,a196a49a198a248a234a41a76a166a115a71a131a105,a198a233a27a41a123a115a184a139
a105a245a107a47,a198a52a252a187a85a142a5a123a24a199a252a202,a208a207a252a187a85a41a47,a198a52a252a187a85a117a127
a123a48a1a92a159,a4a150a164a203a248a252a187a85a86a132a249a75a67,a79a143a198a123a107a47,a27a41a123a176a211a166
a68(Ploytechnique)a210a79a193a229a117a234,a105a159a24a227a30a207,a27a41a33a80a123a166a9a209a243a176a211
a166a68,a183a46a176a211a166a68a15a142a54a25a243a123a56a149,a139a142a245a80a123a166a9,a190a152a128,a27a41a24a199
a33a76a123a106a166a27Poincare a210a52Ploytechniquea123a166a9,Mongea52a145a24a199a85a7a149
a135a73a1a91a86a123a124,a198a123a86a210a119a41a21a135a232a73a243a1a91a222a123a97a126a22,a89a15a52a183a7a89
a123a11a248,a79a36,a27a41a123a115a184a243a135a73a1a91a139a24a199a35a104a123a4a58,a248a234Mongea253a124
a131a105,a132a149a198a243a176a211a123a107a47,a198a139a105a245a166a9,a209a52a243a135a73a1a91a139a35a104a224a26a123
a124,a108a128a243a135a73a1a91a197a76a21a48a131a165,a7a118a116a24a74a123a52a27a41a12a105a24a199a76a106a166
a27Darboux,Darbo uxa52a27a41a41a166a211a123a179a86a159,a196a49a243a198a123a30a207,a198a52a243a41
a5a142a139a105a245a107a47a123a24a199a124,a198a88a157a52a179a86a159,a15a52a174a169a76a166a174a166a211a123a211a159.
a198a123a33a76a123a211a42a52a143a253a21a77a193a88a22,a183a46,a89a143a76a253a52a106a166a169a26a176a62a123a207a123
a24a199a169a26,a25a243a44a234a132a149a199a52a27a169a123,a105a245a124a88a23a89a199a86,a183a46a89a74a124a233a135
1
a73a1a91a98a232a24a154a234a137,a198a162a97a148a23a89a199a86,Darbouxa123a86a89a122a58a158a80,a221a105a234
a105a245a97a238,a2431941 a35,a183a243a220a8a201a76a115a86,a115a21a135a73a1a91a22,a15a89a116a77a193a88.
a89a116a77a193a88a30,a104a108a210a23Darbouxa123a86,a210a46a116Darbouxa123a86a176a62,a24a199a204a7
a123a48a27a52a126a217a196a41a37,a15a210a52a102a126a217a196a41a37a27,a198a126a122a58a158a180a142,a41a122a58a158
a131a164a224,Darbouxa227a135a88a126a123a24a154a52a198a88a126a105a135a73,a183a46,a183a123a49a52a89a135a232
a73,a13a135a232a73a28a7a89a116a245a195,a245a35a106a123a30a127,a89a199a105a135a73a88a21a27a189,a89a52a79a143
a243a245a35a106a123a30a127,a33a139a72a123a211a228a52a105a135a73,a105a135a73a44a49a28,a62,a44a49a198,a44a49
a135a73,a196a49a139a105a245a83a106a123a228a174a44a49a126a116a105a135a73,a51a30,a24a199a105a135a73a15a52a24a199
a42a6,a89a199a42a6a201a163a105a245a106a166a175a11,a88a11a52a199a123a1a91,a164a52a199a123a73a219,a209a201a28
a105a245a97a238,a79a36a52a58a158a139a126a123,Darbouxa123a98a154a52a198a150a139a126a105a135a73,a198a126a217
a196a41a37a27,a98a52a150a139a105a135a73,a79a36,a139a105a245a211
a42,a88a126a105a135a73,a14a143a205a17? a198a15a52a164a7a126a135a232a73a123,a88a126a105a135a73,a198
a126a160a135a73,a126a160a135a73a7a105a135a73a139a122a245a234,a79a143a28a123a77a193a522a145a123a56a45,a196a49
a233a149a220a199a35a106,a253a77a193a123a107a106u,v,a28a7a233ua70a160a135a73,a233va70a160a135a73,a89a176a62
a139a105a245a160a135a73,a13a126a105a135a73a210a59a92a245a234,a98a52a198a88a102a126a105a135a73,a89a52a105a219
a5a123a47a60,Darbouxa52a21a25a24a39a105a135a73a42a123a145a24a124,a24a199a52Darboux,a24a199
a52Frobenius,a198a162a220a199a124a33a0a21a25a89a199a192a220a123,a98a52a127a116a97a126a116a77a193a207a196a123
a30a127,a88a127a119a143a31a143,a198a150a139a126,a15a142a52a132a149a4a58a123a7a248,a198a85a234a105a135a73a131
a128,a120a209a88a195a234,a196a49a198a88a126a105a135a73a234,a183a166a98a89a234,a2431941a35,a183a166a28a243a102
a210a115a89a199a49,a183a105a7a108a143a46,a143a31a143a88a126a105a135a73a17?a196a49a183a210a126a105a135a73a46a27
a6a41Darbouxa196a41a123a211a42,a221a86a128a150a232a41a77a193a88a90a24a74a1a91a123a255a88,a183a102a126
a105a135a73,a79a36a183a122a116a24a199a105a80a123a234a137.
2 a77a193a88a228a253a19a142a123a195a250
a31a143a119a105a135a73a17?a210a52a28a21a25a7a207a196a77a193a123a155,a77a193a52a24a1992a145a123a22a111,a199a243
a202a47a56a45a176a62a522a145a123,a196a49a199a222a62a127a63a24a199a154a52a220a199a35a106,a47a158a210a119a41a107
a106,a98a52a25a243a17,a183a162a210a119a199a219a92a43a41u,v,a79a36a199a123a43a41a522a199a35a106a123a60a106,
a196a49a52a89a199a30a71a38a199a243a154a24a154a139a24a199a35a178a193,a89a199a35a178a193a104a108a105a7a155,a79a143
2
a183a162a150a139a27a6a207a196a133a236a123a67a111,a183a162a144a21a207a196a33a59a92a123a152a134a34,a178a193a89a74a192
a220,a35a178a193a203a77a193a139a33a183a35a123a7a248,a0a143,a92a128a183a35a123a7a248a88a234,a24a189a7a137a219
a143a21a234a137a251a7a118a205a253a47a123a24a74a175a11,a139a24a199a35a178a193,a243a89a199a35a178a193a222a522 a145
a123,a149a52a154a24a154a210a139a142a245a37a222,a15a210a52a35a37a222,a35a37a222a210a52a203a89a199a77a193a35a35
a123a37a222,a79a143a89a199a77a193a52a243Euclid a56a45a176a62,a196a49a183a44a49a89a89a199a37a222a123a159a221.
a143a59a92a229a68,a183a33a149a255a88a159a221a127a1491 a123a37a222,a253a92a160a37a222,a196a49a139a24a87a92a160
a35a37a222,a203a89a74a92a160a35a37a222a18a134a123a139a12a105a24a199a37a222,a183a162a34a189a199a52a82a196a92a160
a123,a0a143a89a199a37a222a183a162a119a41a92a160a27a37a222,a7a213a63a123a52a89a176a210a139a24a199a1a91a25a54
a21a9a234,a79a143a34a38a89a199a77a193a65a178a234a123a155,a92a160a27a37a222a44a49a53a222a18,a15a44a49a53
a5a18,a166a233a155a128,a89a199a77a193a248a234a52a24a1992a145a123a22a111a131a105,a199a164a139a24a199a189a53,a243
a77a193a222a28a52a126a48a53a221,a164a52a203a31a48a53a221,a89a199a221a196a52a105a88a24a248a123,a196a49,a28a7
a189a14a143a248a6a221a196a52a126a48a53,a89a210a52a7a201a77a193a24a199a189a53,a189a53a139a234a131a128,a199a123
a92a160a27a37a222a210a189a234,a92a160a27a37a222a243a89a199a48a53a44a49a53a222a18,a15a44a49a53a5a18,a189
a234a24a199a131a128,a89a199a77a193a15a210a189a53a234,a89a52a105a173a7a123a24a199a10a39,a175a108a35a139a123a144
a52a24a199a110a82,a98a52a89a52a24a199a105a173a7a123a10a39,Mobiusa52a121a41a149a76a123a1a91a166a27.
a79a143a28a7a189a53,Mobiusa21a25a139a74a77a193a88a21a189a53,a89a104a108a52a105a139a63a135a123a24a71a47
a60,a28a162a76a27a209a127a119a123a89a199a67a111,a210a52a252a24a60a146,a28a178a199a221a24a87a203a229a117a123a155,
a210a122a116a196a162a123Mobius a77a193,a199a150a27a6a189a53,a89a52a1a91a222a105a139a63a135a123a24a71a47
a60,a0a143a183a162a34a189a77a193a46a178a104a189a234a24a199a48a53,a139a234a89a248a189a53a123a77a193a131a128,a1
a91a60a89a52a14a143a248a123a17?a132a92a160a27a37a222e3,a217a165e3 a52a2433a145a56a45,a28a21a25a139a24a71
a47a34,a210a52a128a28a92a213a255a88a77a193a88a234,a28a24a189a7a188a126a77a193a222a123a92a160a35a37a222,a183
a119a89a199a92a160a35a37a222a143e1,a89a248a183a210a139a234a24a199a41a37,a199a139a234a145a24a199a92a160a37a222a90
a145a174a199a92a160a37a222,a152a42a56a45a52a189a53a123,a145a19a199a92a160a37a222e2 = e3 ×e1a15a210a113a92
a104a189a234,a196a49a183a210a139a199a92a160a41a37,a92a160a41a37a210a52a174a199a92a160a37a222a149a76a24a189a123
a39a147,a52a144a35a18a134a123,a143a31a143a92a160a41a37a243a1a91a123a207a196a131a165a52a89a143a173a7? a210a52
a79a143a1a91a52a202a226a228a196a107a207a196a56a45a243a228a196a131a5a88a35a123a1a91a117a159,a13a89a228a196a107
a210a52a41a37a196a196a123a56a45,a79a143a52a139a24a199a63a37a144a139a24a199a228a196a178a24a199a41a37a35a143a217
a199a123a41a37,a150a149a92a14a123a92a160a41a37a203a89a199a228a196a107a123a195a163a196a24a24a233a97,a88a98a52a24
a24a233a97,a13a37a233a97a226a209a95a192a90a24a35a123a117a159,a196a49a228a196a107a105a7a155,a79a143a56a89a123
3
a228a196a88a127a119a243a137a219a123a60a89a131a5a152a91a44a49a255a174,a13a234a41a37a131a128,a210a44a49a255a174a234.
a41a37a210a52a37a222a234,a13a37a222a24a196a52a1393a199a73a222a123a37a222,a13a154a24a199a73a222a52a60a106,a210
a44a49a178a199a135a73,a28,a62a31a143a123,a37a222a139a28,a62a123a228a174,a15a139a135a73a123a228a174,a243a236
a171a63a66a5,a164a44a49a139a232a73a123a228a174,a196a49a183a25a243a210a44a49a135a73,a183a207a196a77a193a123
a30a127,a88a144a24a199a41a37,a0a143a243a77a193a123a154a154,a89a248a123a41a37a139a245a232a17? a34a38a28a65
a122e1a123a155,a51a30a89a199a77a193a52a189a53a123,a89a199a41a37a210a113a92a189a234,e1a52a31a143a17? e1a52
a89a199a77a193a243a89a24a154a123a92a160a35a37a222,a0a143a89a199a77a193a139a245a232a92a160a35a37a222a17? a154a154
a139a24a87a243a35a178a193a222a62a127a149a92a160a37a222,a13a77a193a522a145a123,a196a49a199a162a196a196a123a56a45
a523a145a22a111,a89a52a79a143a89a199a154a52a243a77a193a222a35a196,a522a145a123,a25a243a243a154a189a234a131a128,
a92a160a35a37a222a44a49a119a248a199a221a24a87,a196a24a199a203a177,a196a49a199a52a145a28a24a145,a523a145,a89
a1993a145a56a45a58a158a7a155,a183a46a25a243a34a19a222,a28a162a7a234a137a135a232a73a221a86a234a137a203a135a232
a73a5a86a123a106a166a221a86a243a106a166a165a123a97a126,a89a199a60a89a52a33a59a92a123,a51a30a52a33a139a126a123.
a196a49a183a139a24a1993a145a56a45,a132a149a154a24a154a139a199a203a177,a25a243a139a199a214a9a119a41a203a45,a221a86
a203a177a45,a45a52bundle,a196a49a28a7a207a196a77a193a123a1a91a117a159,a126a89a199a137a219a123a48a27,a24
a189a7a255a88a199a123a203a45,a255a88a203a45a234a131a128,a24a35a209a59a92a234,a79a143a24a35a209a52a37a222,
a13a52a37a222a123a155,a199a139a73a222,a210a44a49a135a73,a210a44a49a126a83a106a221a86a135a73a123a228a174,a183
a162a52a243a255a88a135a232a73,a183a162a34a189a141a116a31a143a60a106a209a44a49a135a73,a183a119a243a89a199a77a193a222
a123a154a143x,a0a143dx a52a24a199a37a222,a210a52a44a198a154a203a248a89a199a154a123a37a222,x a52u,v a123a60
a106,a13u,va52a77a193a222a123a219a92a43a41,a196a49a28a44a49a85a241dx,a34a38x a33a155a243a77a193a222,
a0a143dxa24a189a52e1 a166e2a123a34a117a28a92,a196a49a243a89a199a143a48,a183a210a223a73a188a126a105a135a73a123
a10a39,a34a19a222dxa52a24a199a37a222a138a123a24a39a135a73a42,a196a49a199a52e1 a166e2 a123a34a117a28a92,
a199a123a28a92a248a106a52a24a39a135a73a42,a196a49dxa44a49a85a143
dx = ω1e1 + ω2e2,(5.1)
(dx,dx) a210a52a183a162a77a193a123a169a121a221a222,a79a143e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49
ω21 + ω22 = ds2 (5.2)
a210a52a169a121a221a222,a152a42a89a199a56a249a234,a89a233a149a202a47a89a135a73a1a91a59a92a245a234,a79
a143a202a47a135a73a1a91,a169a121a221a222a7a85a196gijdxidxj,a89a52a79a143a243a35a56a45a176a196a188a126
4
a123a43a41a52a127a63a123Ca rtesiana43a41,a199a88a24a189a18a134,a15a88a24a189a52a92a160,dx a127
a149ω1e1 + ω2e2,a98a52a183a162a105a135a73a139a199a228a253a123a117a159,a210a52a242a126a24a39a123a155,a199a127
a1490,a89a210a52a202a47a128a123a160a135a73a44a49a52a98a166a123a30a71,a24a248a123,a15a210a52a122a116a123a160a135
a73a166a135a73a123a39a147a195a7,a196a49a28a210a178da126a116dxa222a62,a24a189a127a1490,a28a178a141a30a48a13
a123a155,a210a122a116d(ω1e1) + d(ω2e2),a213a63a104a105a135a73a3a193a139a24a199a24a39a79a42a123a155,a135
a73a145a19a199a79a6a7a149a82,a15a13a211a131,a44a49a122a116
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(5.3)
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(5.4)
a183a204a243a5a193a201a241ω12,ω13,ω23,a17a108a126a135a232a73a234,a196a49a44a49a178(xe1e2,e3)a123a135
a73a44a143e1,e2,e3a123a34a117a28a92,a89a199a34a117a28a92a178deia85a196ωijej,a25a243a183a126a135a73a1
a91a202a47a123a110a82,a34a38a139a24a199a141a106a7a173a133a123a155,a210a44a43a35a28,i,j,ka441a1163,a28
a178dei a85a196ωijej,ωija123a1a91a63a66a105a210a23,a28a25a243a139a24a28a41a37,a89a28a41a37a203a24
a28a107a106a139a7a248,a13a233a149a89a24a28a41a37,a210a139a24a199a249a163a41a37,a89a199a249a163a41a37a203a198
a117a41a37a123a7a248a210a52ωij,a89a7a248a52a132a24a39a135a73a42a117a44a43a123,a79a36a210a139
dei = ωijej,(5.5)
a89a28a48a199a42a105a7a155,a199a210a44a43a220a199a249a163a41a37a144a35a123a7a248,a243a89a199a60a89a131a5,
a135a73a1a91a203a197a166a88a76a24a248,a197a166a128a128a35a106a52a30a45,a196a49a24a199a41a37a203a248a30a45a243
a35a196,a79a36a28a114a199a41a37a144a139a24a199a35a106,a209a52a30a45ta123a60a106,a25a243a183a162a52a24a199a77
a193,a154a154a139a142a245a41a37,a196a49a183a89a41a37a123a107a106a523,a89a52a79a143a139a35a193a123a219a92a43
a41,a145a139a35a37a222a243a178a193a176a62a35a166a123a43a41,a196a49a183a25a243a89a199a7a35a106a523,a164a79
a143E a89a56a45a523a145a123,a7a35a106a176a234,a196a49a89a52a139a198a79a38a122a105a135a73a139a72,a183a162a46
a82dei a85a196a48a199(5.5),ωija233a149i,ja52a39a233a193a123,a89a52a79a143a183a123a41a37a52a92a160a41a37,
a253a79a143(ei,ej) = δij,a196a49a199a52a39a233a193a123,a79a36ωija34a19a222a105a59a92,a28a178(ωij)a85
a241a117,a199a52a24a199a48a106,a89a199a48a106a52a39a233a193a123,a196a49a243a233a110a34a123ωa127a1490,a217a155
a123a233a248a233a110a34a52a39a233a193a123,a79a36a34a19a222a144a1393a199a24a39a135a73a42,ω12,ω13,ω23.
a183a46a183a222a39a121a210a234ω12a132dω1,dω2a123a48a199(5.3)a113a92a104a189,a89a52a24a199a173a7a123a189
5
a174,a89a52a38a122Levi-Civitaa241a214a123a173a7a189a174,a183a25a243a178a48a199(5.5)a70a105a135a73,a79
a143d(dei) = 0,a196a49a141a30a123a155,a183a210a122a70dωij,a136a42a122a116a123a52
dωij = ωik ∧ωkj,(5.6)
a79a36a89a74ωa131a45a139a105a59a92a123a7a248,a59a92a122a88a122a234,a79a143a31a143a17?a79a143a233
a149dωij = ωik ∧ωkj,i,ja52a88a35a127a123,a152a42a35a127a234a123a155,ωii a520,a89a52a79a143ωa52
a39a233a193a123,a196a49a28a82i negationslash= j,a152a42ka127a149i,a11ωii = 0; ka7a127a149j,ωjj = 0,a196a49k
a88a127a149i,a88a127a149j,a79a143a183a162a52a2433a145a56a45,ka144a139a24a199a44a21a117,a79a36a89a199a23a248
a105a0a219a123a48a199a42,a199a123a141a30a144a139a24a49,a183a222a39a178a199a85a5a117a234,a210a52
dω12 = ω13 ∧ω32,(5.7)
dω13 = ω12 ∧ω23,(5.8)
dω23 = ω21 ∧ω13,(5.9)
a131a217a52a122a116dω12 = ω13 ∧ω32a89a199a218a42,a98a52ω a52a39a233a193a123,a196a49a210a122a116
dω12 =?ω13 ∧ω23,(5.10)
a13ω13,ω23a209a52ω1,ω2a123a34a117a28a92:
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(5.11)
a89a166a28a210a122a116a5a193a89a199a218a42:
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(5.12)
Ka52Gaussa77a71,Gaussa77a71a210a52K = ac?b2,a89a199a218a42a88a122a234,a104a35Gaussa88
a52a89a248a122a116a89a199a218a42,a52a126a107a123a48a27a122a116a218a42,a199a119a41Theorem Egregium,
a126a165a169a89,a199a52a24a199a219a201a123a189a174,a201a123a189a174,a28a251a251a23a23a199,a199a52a105a201a123,a79
a143a183a162a123Ea52a92a160a37a222a45,a199a89a199a203a177a45a233a149a77a193M a139a24a199a61a107,a233a149a203a177
a45,a139a89a199a92a160a37a222,a183a82a199a123a198a154a210a52a199a123a61a107,a79a36a183a162a25a243a123a1a91a7
a44a3a10a39a222a7a118a133a236a234,a210a52a128,a88a144a52a139a24a199a77a193a221a86a139a24a199a77a34,a25a243
6
a139a220a199a56a45,a139Ea89a199a56a45a90a77a193M,a47a34a222,a139a234a77a193M,a108a128a139a132a196a139a199
a123a92a160a37a222a196a196a123a56a45a52E,a199a52a77a193a123a203a177a45,a25a243a47a158a119a89a199a77a193a52a142
a56a45,a199a52a243a142a222a123a24a199a56a45,a196a49a210a139a24a1993a145a123a203a177a45,a199a123a142a56a45a52a183
a162a123a77a193,a243a89a199a60a89a131a5,a183a162a139a1a199a24a39a135a73a42,a243a56a45a176a62a139a24a39a135
a73a42ω1,ω2,a108a128a139ω12,ω13,ω23,a24a2255a199a24a39a135a73a42,a217a165ω13,ω23a52ω1,ω2a123
a34a117a28a92(a218a42(5.11)),a199a44a43a77a193a123a1a91a117a159,a183a162a139
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (5.13)
a145a19a228a253a42 II = (?dx,de3) = ω1 ∧ω13 + ω2 ∧ω23 = aω2 + 2bω1ω2 + cω22.
(5.14)
a15a210a52ω1∧ω13 a28a222ω2∧ω23a52a77a193a196a162a123a145a19a228a253a42,a145a24a228a253a42a52ω21+ω22,
a52a220a199a35a106ω1,ω2a123a178a48a90.a145a19a228a253a42a233a149a145a24a228a253a42a139a1a110a138,a1a110a138a123
a83a106a90a24a196a119a41a165a77a71H = a+c2,a1a110a138a123a232a210a52Gaussa77a71,a196a49Gaussa77
a71a52ac?b2,a13a183a162a25a243a139a24a199a219a201a123a218a42a210a52dω12 =?Kω1 ∧ω2,a243a89
a199a218a42a141a30,ω1 ∧ω2 a210a52a77a193a123a193a232a195a163,a199a104a108a144a203a77a193a123a221a222a139a7,a166
a199a123a160a152a150a139a7a248,a144a203a77a193a123a145a24a228a253a42a139a7,a243a199a123a38a30,ω12 a52a183a162
a123Levi-Civitaa201a100,a132a149a141a30a132a145a24a228a253a42a113a92a104a189,a196a49a38a30a15a52a144a166a145
a24a228a253a42a139a7,a79a36a183a162a127a119Gaussa77a71a144a203a145a24a228a253a42a139a7,a89a52Gaussa58
a158a122a63a123a136a42.
3 Gauss-Bonneta218a42
a183a162a44a49a132dω12 =?Kω1 ∧ ω2 a122a241a117Gaussa139a214a123a89a199a189a174,a183a25a243a128,
Gauss-Bonnet a218a42a15a193a47a210a132a89a199a218a42a122a241a117,a0a143,a104a35a183a15a89a143a46,a79
a143a34a38a28a139a24a199a85a20a123a77a193,a139a234a89a199Gauss a77a71a198a49a193a232a195a163,a28a104a108a105
a7a108a143a175a199a123a232a73a52a31a143,a139a24a199a193a232a195a163a198a49a24a199a60a106(Gaussa77a71),a178
a199a243a77a193a222a232a24a40a122a241a123a106a52a88a52a203a89a199a77a193a123a1a91a139a7?a0a143,a89a199a210
a52Gauss-Bonneta218a42,Ma52a85a20a123a189a53a77a193,a11
integraldisplay integraldisplay
KdA = 12piχ(M),(5.15)
7
a243a183a162a89a199a218a42a165,a183a162a178ω1 ∧ ω2 a119a193a232a195a163,a183a162a15a44a49a119a199a41dA,
KdAa210a52a89a199a232a73,a28a193a47a210a46,a183a162a243a85a20a123a77a193,a196a49a152a42a139a24a199a232
a73,a199a44a49a85a196da31a143a123,a0a143a199a123a232a73a243a97a148a52a243a77a193a123a30a142a222,a44a49a35
a143a24a199a30a142a232a73,a89a52a183a13a41a89a123Stokesa189a174a204a7a123a92a73,a196a49a204a88a204a132
a89a199a122a116Gaussa77a71a123a232a73a520,a89a23a108a52a70a216a123,Gaussa77a71a232a73a88a204a127
a1490,a89a52a79a143a28a82a24a1992a145a123a69a193,a199a44a49a82a24a196a123a221a222a38a199a123Gaussa77a71
a127a1491,a0a143a243a77a193a222a123a232a73a104a108a88a204a127a1490,a196a49a89a199a119a174a52a70a216a123,a104
a35a79a36a183a46a70a243a31a143a143a48,a70a243a183a162a89a199a218a42dω12 =?KdAa88a52a243a77a193a222,
a52a243a203a177a45a222a62,a89a199a77a193a52a243a203a177a45a222,a196a49a183a137a7a178a89a199a77a193a10a176a116
a203a177a45a222a62,a10a176a52a31a143a63a135a17? a210a52a243a154a24a154a28a82a89a199a154a42a143a198a154a123a92
a160a35a37a222,a196a49a166a233a155a128,a183a162a24a196a119a89a199a143a37a222a156,a79a36a28a243a24a1992a145a22
a111a222a193a7a82a24a199a37a222a156,a89a199a37a222a52a92a160a37a222a203a77a193a35a35,a253a143a92a160a37a222
a156,a89a52a88a52a44a21? a177a34a127a201a28a24a199a77a193,a28a21a88a21a154a154a141a189a24a199a92a160a35
a37a222,a221a86a35a37a222,a38a122a199a52a203a153a123,a4a150a52a44a49a135a73a123,a89a248a123a37a222a156a88
a24a189a139,a89a52a24a199a95a192a123a189a174,a114a199a123a89a24a227a123a21a48a52a77a193a88,a221a86a52a114
a14a77a193a88a203a77a193a123a221a222a89a171a171a123a7a248a243a139a88a245100a35a3a52a204a7a123a175a11,a0
a199a30a127a33a204a7a123a236a147a52a21Mathematische Annalena22,a92a45a142a106a166a123a165a101a52
a243a121a41,a21Mathematische Annalena22a222a62a7a149a89a199a175a11a139a105a245a169a57,a0a30
a127,Einsteina52a21Mathematische Annalena22a123a24a199a31a246,a89a199a175a11a113a92a180a56
a249a234a137a7a24a74a30a45,a0a143a217a165a139a24a199a79a163a52a34a38a28a139a24a199a85a20a123a77a193,a28
a82a154a154a123a63a37a189a53a123a92a160a27a37a222,a253a82a154a154a104a189a123a92a160a27a37a222,a178a199a110a243
a116a92a160a69a222a86,a89a199a47a158a119a41Gauss a110a243,a196a49Gaussa110a243a178a28a123a77a193a110a243
a116a92a160a69a222a86,a89a248a210a178a24a1992a145a77a193a110a243a116a12a24a1992a145a77a193,a220a199a209a52a189a53
a123,a149a52a89a176a210a139a24a199a95a192a88a35a42,a119a41Mapping Degree,a253a110a243a123a39a106,a243
a178a193a123a60a111,a28a139a24a30a77a34,a196a49a92a160a69a210a52a92a160a203,a0a143a28a178a89a30a85a20a123
a77a34a110a243a116a92a160a203a222a62,a196a190a199a221a234a245a232a87a17?a89a210a52Mapping Degree(a110a243
a123a39a106),a89a199a189a66a243a176a145,a4a150a522a145a210a88a89a143a59a92a234,a23a108,a183a162a178a77a193
a110a243a116a92a160a69a222a62,a108a128a82a92a160a69a222a62a123a193a232a195a163,a89a210a52Gauss a77a71,a196
a49a89a199a110a243a210a127a149Gaussa77a71a123a232a73,a178a89a199a233a193a56a45a123a193a232a195a163a126a110a243a110
8
a195a117a131a128a90a198a117a77a193a123a193a232a195a163a24a248,a253a143a70a199a123a114a199a123a232a73,a210a122a116a110
a243a123Degree,a44a49a121a210a89a199Degree a243a44a49a189a53a123a77a193a123a60a89a5,a199a52a77a193
a123Eulera106a123 12pi,a105a44a234a123,a183a76a150a150a139a30a45a178a89a199a113a92a89a113,a183a46a183a5a39
a242a45a251a89a89,a183a25a243a76a151a89a89a89a199a121a210a52a14a143a248a123,a23a108a183a162a46a139a228a253a218
a42dω12 =?KdA,a183a162a164a7a178a77a193a56a116Ea222a86,a196a49a7a189a24a199a92a160a37a222a156,
a89a88a24a189a44a21,a89a248a123a92a160a37a222a156a7a139a73a154,a15a210a52a128a152a42a28a227a142a114a14a92a160
a37a222a156a139a73a154a123a155,a199a0a52a44a21a123,a89a15a137a7a121a210,a139a234a73a154a131a128,a210a23a89a199
a73a154a123a117a159,a7a152a128,a28a23a23a89a74a92a160a37a222a156,a152a89a1a199a67a195a67a81a196,a89a52a37
a222a156a123a73a154a123a44a21a117,a44a21a37a222a156a139a142a245a88a51a123a48a53,a98a52a243a73a154a139a30a127a89
a199a37a222a53a105a18,a15a139a30a127a37a222a53a176a18,a15a139a30a127a210a54a145a174a199a67,a199a52a203a24a24
a118a77a34a35a35,a196a49a255a88a37a222a156a123a73a154a123a117a159a52a58a158a139a63a135a123a175a11,a0a143a183
a243a127a91a77a193a24a189a139a24a199,a63a37a37a88a144a24a199,a13a52a139a142a245a139a73a154a123a92a160a37a222a156.
a243a89a199a60a111a131a5,a28a7a178a77a193a35a196a66a76,a35a196a24a199a196a162complex(a133a111),a28a11
a189a199a243a186a154,a79a143a196a162a123complex,a210a52a128,a139a74a186a154,a139a74a30,a139a74a193,a243a186
a154,a189a24a199a127a91a123a37a222,a0a143a145a19a90a178a89a199a37a222a156a210a116a30,a82a24a199a30a123a155,a152
a42a37a222a156a46a178a243a186a154a104a189a234,a0a143a105a142a52a23a241a117a44a49a178a199a210a116a114a199a123a30,a25
a243a52a193a234,a34a38a52a193a123a155,a7a48a128a139a199a174a110a111,a25a243a37a222a156a46a178a243a30a222a209a189
a234,a52a88a52a21a234a210a159a116a199a123a19a92,a89a210a88a210a23a234,a104a108a89a52a24a189a44a49a123,a34
a38a28a227a142a199a139a73a154,a7a48a128a52a24a199a174a110a111,a0a143a28a210a243a174a110a111a165a45a82a24a154,
a108a128a44a89a154a203a116a30a222,a0a143a89a199a37a222a156a210a178a199a210a116a19a92,a98a52a89a248a189a123a37a222
a156a243a165a45a123a0a154a24a189a52a73a154,a79a143a165a45a123a0a154a150a27a6a189a24a199a44a49a203a153a123a37
a222a156,a38a122a154a154a144a139a24a199a37a222,a243a89a24a154a210a150a27a6a189,a196a49a166a98a89a199a174a132a44
a49a121a210,a34a38a28a227a142a89a37a222a156a139a73a154,a89a248a123a37a222a156a52a63a243a123,a202a47a24a189a139
a73a154,a243a143a69a222,a7a253a90a123a155,a89a90a203a143a69a35a35a123a30a127,a199a123a48a53a52a199a37a222
a156,a24a189a139a24a154a150a139a90,a150a232a139a24a154,a13a89a24a154a210a105a133a236a234,a0a143a89a171a248a6a154
a123a207a196a52a105a139a63a135,a15a105a7a155a123a175a11,a75a47a37a222a156a243a73a154a123a117a159,a105a59a92a123
a24a199a48a27a52a183a162a44a49a243a73a154a189a24a199a114a106,a89a24a196a210a119a41a199a123a141a41a213
I(s) = lim 12pi
integraldisplay
ω1,(5.16)
9
a210a52a128,a165a45a139a73a154,a73a154a142a163a123a37a222a156a52a113a92a104a189a123,a196a49a28a243a89a154a152a199
a66a203a177,a0a143a203a177a123a30a142a37a222a156a46a178a139a234,a196a49a28a243a89a24a154a178a199a110a243a116a24a199
a203a177,a79a143a219a92a123a30a127,a28a34a189a52Euclida1a91,a34a189a126a178a113a34,a28a210a178a199a110a116
a203a177,a0a143a119a203a177a52a245a232a39a17? a89a210a52a37a222a156a123a141a41,a5a39a183a121a210a34a38a28a139
a234a89a74,a28a210a82a92a160a37a222a123a37a222a156,a63a37a178a89a199a77a193a10a176a116Ea176a62,a253a10a176
a116a203a177a45a176a62,a199a210a35a89a199a77a193a143a139a43a154a123,a17a139a43a154,a145a139a73a154,a89a199a73a154
a65a222a86a210a52a43a154,a0a143a89a74a43a154a123a141a41a28a243a24a229a210a52Euler a158a106,a210a127a149a89
a199a232a73,a253a89a199a232a73a139a63a66,a199a52Eulera158a106,a183a46a135a232a73a139a24a199a173a7a123a97a126
a52a243a133a35a60a106a88,a28a7a89a106,a33a139a63a135a123a106a210a52a133a106,a105a110a99,a183a23a165a41a106a166
a36,a165a41a124a212a34a19a234,a165a41a124a150a139a133a106,a133a106a7a155a122a88a122a234,a183a7a89a24a154a133
a35a60a106,a183a7a121a83a106a228a253a189a174,a133a35a60a106a131a128,a127a91a83a106a48a199a42a209a139a137,a89a52
a88a122a234a123a24a199a136a42,a89a199a136a42,a104a35Eulera88a204a121,a105a245a124a209a121a88a241a117,Gauss
a21a121a210,a198a52a163a83a33a149a76a123a106a166a27.
10
a145a24a89 a77a193a88(a174)
a9Gauss-Bonneta218a42(a153)
2001a3511a21930a134
1 a135a232a73a243a133a35a60a106a88a165a97a126a59a144
a183a164a97a148a242a89a220a39,a89a220a39a183a139a199a15a153,a189a247a89a24a154a133a35a60a106a88,a79a143a243a106
a166a165,a105a7a155a123a24a71a47a34,a51a30a243a106a166a36a222a15a52a58a158a7a155a123a24a71a47a60,a210a52
a139a133a106,a89a199a133a106a38a122a106a166a59a92,a133a60a106a139a142a245a164a224,a139a63a135a123a117a159,a79a36,
a89a38a122a89a74a60a106a243a97a126a222a1a47a139a126a255,a196a49,a183a189a247a89a24a89,a7a152a128,a133a35a60
a106a139a24a199a105a173a7a123a117a159,a127a63a123a83a106a48a199a243a133a35a60a106a131a165a24a189a139a137,a89a52a24
a199a88a122a234a123a47a60,a79a143a88a11a28a14a143a248a85a24a199a48a199,a28a7a52a227a142a137a52a133a106a123a155,
a199a24a189a139a137,a190a152,x2 + 1 = 0,a0a143a199a139a199a137a210a52√?1,a196a49√?1a210a89a143a248
a6a139a126a255,a88a98a152a36,a133a106a203a34a106a24a248,a44a49a28a62,a139a51a248a123a117a159,a196a49,a199
a44a49a228a174,a51a30a199a221a57a234a142a245a97a238a52a34a106a88a21a221a57a123,a183a46a183a123a49a243a44a199a165
a24a189a204a139a199a56a105,a243a56a105a123a30a127,a183a46a73a220a39a89a133a35a60a106,a183a189a247a89,a24a199a52
a183a166a98a89a123a83a106a123a228a253a189a174,a210a52a128a127a63a123a83a106a123a48a199a243a133a106a173a165a24a189a139
a137,a89a199a52a105a10a121a210a123,a137a7a106a166a222a99a123a10a39,a7a48a128,a149a76a106a166a27a152Euler,
a198a46a27a6a121a210,a98a150a139a21a196a213,a183a46Gauss a52a183a162a163a83a33a149a76a123a106a166a27,a198
a105a35a52a123a30a127a210a139a24a199a121a210,a15a210a52a133a106a137a7a24a74a1a91a123a117a159,a88a113a92a52a83
a106a123a175a11,a183a189a247a5a39a89a133a106a123a30a127a121a210a89a199a189a174;a51a30,a133a35a60a106a33a204a7
a123a24a199a189a174a52Picarda189a174,a210a52a128,a34a38a233a149a24a199a133a35a60a106,a82a199a123a60a106a138a243
a133a178a193a176a62a196a82a123a160a152,a199a178a114a199a133a178a193a209a152a212a234,a217a165a15a142a44a49a86a171a24a154,
a220a154,a89a52a88a122a234a123,a210a52a128,a60a106a152a42a52a24a199a92a24a60a106a123a155,a199a73a89a122a58a158
a131a254a225,a44a49a128a139a88a245a178a178a193a209a152a212a234,a139a63a135a123a24a71a47a60a52a89a199a189a174a52a133
a35a60a106a176a88a123a189a174,a44a49a188a126a183a162a25a243a7a89a123Gauss-Bonneta218a42a117a121a210,a89
a128a210a23a229a117a150a139a7a248a123a24a74a48a27a203a10a39,a136a42a52a139a7a248a123,a89a52a106a166a222a58a158
a7a155,a139a63a135a123a175a11.
1
2 a7a149a166a243a123a7a196a117
a89a199a49a79a136a101a234,a28a162a243a89a199a49a85a199a231a183,a33a80a52a7a196,a28a21a234a7a3a73a116a24
a199a175a11,a89a52a205a7a155a123,a183a46a28a162a209a52a76a166a9,a76a166a9a73a176a127a115a184a33a128a123a24
a227,a49a128a116a246a204a222a86,a253a38a243a166a68,a243a166a98a92a160a176a62,a33a7a155a123a24a189a7a7a196.
a88a7a52a127a144a19a119a28a41a31a143,a28a242a41a31a143,a89a199a33a160,a7a7a196,a7a7a3a21a73a175
a11,a7a7a3a21a234a73a133a7a3a73a123a175a11,a0a143,a104a108a28a73a123a175a11a88a24a189a92a55,a28a246
a30a15a88a24a189a21a234a122a116a73a153,a88a44,a28a165a45a178a44a24a74a107a52,a178a44a24a74a70a216,a44a49
a38a122a28a123a166a175a93a116a159a90,a13a38a122a28a93a116a159a90a123a210a52a7a178a44a89a248a123a67a153,a196a49
a183a243a185a76a27a7a7a196,a245a24a154a143a89a229a117,a28a162a4a150a7a21a234a28a132a24a199a76a14,a144a35
a231a183a73a175a11,a221a86a62a68a19a68a105a144a19,a51a166a117a41a231a183,a89a52a105a139a80a255a123,a7a3a7
a178a106a166a46a24a46,a221a86a233a127a63a123a166a175,a28a7a3a139a199a135a46,a250a122a139a199a31a143a248a123a217
a196,a233a149a28,a233a149a89a199a166a175a123a127a35a44a49a15a28,a51a30a28a233a166a175a123a21a197a15a44a49a15
a28,a196a49a89a52a105a138a122a213a63a123a24a71a47a60,a230a130a28a162a31a68a24a5a89a199a44a21a117.
3 Gauss-Bonneta218a42a123a121a210
a222a39,Gauss-Bonneta218a42a183a150a139a121a210a92,a196a49a183a11a178a121a210a128a92a234,a183a222a39a89
a123Gauss-Bo nneta218a42a210a52,a34a38a243a56a45a176a62a139a24a199a77a193,a199a52a24a199a114a199a123
a77a193,a63a37a34a38a89a199a77a193a52a189a53a123,a253a199a123a27a34a139a24a189a123a48a53,a149a52a89a248a6,
Gaussa77a71Ka210a52a77a193a222a123a24a199a60a106,a183a44a49a178a89a199a60a106a233a149a77a193a222a123a193a232
a221a222a70a232a73,a89a199a232a73a52a24a1992a173a232a73,a70a199a232a73a131a128,a136a42a89a199a232a73a127a149a24
a199a158a106(2pi)a198a49a77a193a123Euler a43a117a106,a253
integraldisplay integraldisplay
KdA = 2piχ(M) (6.1)
Eulera43a117a106a210a52a178a77a193a35a196a66a76a131a128,a55a149a24a154a7a108a123a30a71,a178a199a35a113a131
a128,a217a186a154a199a106?a30a123a199a106+a193a123a199a106,a89a2483a199a106a123a116a139a123a90a210a119a41a89a199a77
a193a123Eulera43a117a106,a104a77a193a52a69a193a123a155,a199a123Eulera43a117a106= 2,a152a42a199a52a199a162
a193,a199a123Eulera43a117a106a520,a28a162a44a49a65a24a65,a210a21a122a116a89a199.a152a42a77a193a52a199a189
2
a53a77a193,a89a52a199a123a141a24a123a95a192a88a35a42,a24a196a89a229a117,a34a38a69a222a28a1a199a162,a162a123
a199a106a210a203Eulera43a117a106a139a199a7a248,a89a162a123a199a106a202a47a119a77a193a123a90a194(genus),a89
a52a77a193a33a173a7a123a95a192a88a35a42,a139a63a135a123a52,a89a77a193a123a117a159,a77a193a222a62a60a106a123
a117a159a203a90a194a139a183a35a123a7a248,a196a49a90a194a52a95a192a88a35a42,a199a107a47a116a77a193a123a1a91a117
a159a90a137a219a117a159,a139a58a158a131a173a7a123a107a47,a196a49a114a199a89a74a7a248a52a105a253a163a123,a35a104
a253a163a123,a79a36,a15a52a58a158a7a155,a58a158a139a63a135a123,a183a222a39a121a210Gauss-Bonneta218
a42,a33a7a155a123a218a42a210a52
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(6.2)
a183a25a243a173a133a24a40,a7a207a196a77a193a88a123a155,a24a189a7a207a196a77a193a222a123a41a37,a34a38a82a89
a199a41a37,a38a41a37a1233a199a92a160a37a222a144a35a18a134,a63a37a183a162a34a189a199a52a199a141a67a248,a253
a243a220a199a131a45a160a10a24a199a141a67a221a86a38a67,a183a162a34a38a52a141a67a248,a0a143,a233a149a89a248a6
a41a37,a34a38a28a127a119a145a24a199a37a222a131a128,a217a199a220a199a37a222a210a104a189a234,a79a143a183a162a34a189
a145a174a199a52a77a193a123a92a160a27a37a222,a0a143a145a24a199,a145a174a199a189a234a123a155,a145a19a199a15a210a189
a234,a47a34a222,a183a89a52a24a199a92a160a41a37,a51a30a52a141a67a248(a141a67a41a37),a89a210a113a92a189a234.
a196a49a233a149a243a24a199a154a123a196a139a89a171a248a6a123a41a37,a24a225a89a171a41a37a139a92a107a106a248(one
parameter family),a52a202a226a234a24a199a35a106,a77a193a522a145a123,a242a28a222a89a154a123a41a37a139
a24a199a107a106,a196a49a77a193a196a139a41a37a52a24a1993a145a123a56a45,3a145a56a45a139xa89a199a186a154,a189a199
a243a77a193a123a160a152,a199a86a171a220a199a145,a108a128a242a82a24a199a35a34a48a53,a145a139a24a199a145,a79a143a35
a34a243a35a193a176a62a44a49a221,a196a49a145a245a234a24a145,a89a248a6a210a122a116a196a139a41a37a248a196a196a56a45
a1233a145a123a117a159,a139a234a41a37a248,a139a31a143a80a255a17? a79a143a139a234a37a222,a28a44a49a126a218a42a117
a44a43a241a117,a37a222a139a73a222,a89a73a222a210a139a106,a183a162a180a106a166a33a7a155a123a7a139a106,a28a7
a139a106a123a155,a195a85a52a239a104a123,a63a37a97a126a123a30a127a28a44a49a10a137a116a123a209a52a106,a243a236a171
a63a66a5a143a31a143a135a232a73a7a155? a183a46a106a166a204a7a123a248a123a52a207a196a60a106,a207a196a220a199a248a58
a123a7a248,a25a243a89a7a248a17,a60a106a88a80a180a234,a196a49a135a232a73a52a178a89a199a7a248a34a117a154,a79
a36a44a49a126a83a106,a37a222a44a49a28,a252a199a106a248a117a198,a196a49a135a232a73a204a7a123a196a210a52a178a56
a45a123a174a88,a178a60a106a123a174a88a34a117a154,a83a106a154,a139a234a83a106a49a128,a28a210a44a49a174,a196a49
a210a139a126,a79a36a15a173a7,a0a143a139a234a41a37a196a122a116a123a137a219a123a47a34a52a31a143a17? a183a178a89
3
a199a41a37a119a41xe1e2e3,
E = {xe1e2e3|Ma189a53 e3a52a27a37a222,e1 mbox4#%} (6.3)
e3a52a92a160a123a27a37a222,x,e1,e2,e3 a209a52a37a222,a196a49a199a162a123a135a73a15a52a24a199a37a222,a135
a73a131a128a52a24a39a135a73a42a37a222a138,a79a36,a199a162a44a49a44a143e1,e2,e3a123a24a199a34a117a28a92.
a183a178dxa44a143a34a117a28a92,a122a116a123a248a106a183a119a41ω1,ω2,
dx = ω1e1 + ω2e2,(6.4)
ω1 ∧ ω2a210a52a77a193a123a193a232a221a222,a52a24a1992a39a135a73a42,a199a104a108a44a49a126a117a41a199a173a232
a73a123a232a73a60a106(integral),a196a49a178a199a232a73a123a155,a210a122a116a89a199a77a193a123a193a232,a183a89
a123a7a149a77a193a123a174a88a123a89a74a136a42,a28a243a135a73a1a91a86a222a73a88a116,a152a42a28a88a21a113a92
a122a73,a88a21a113a92a195a123a155,a150a139a7a248,a79a143a89a74a19a142a76a150a52a202a47a135a73a1a91a44a49
a89a24a199a219,a183a89a24a220a39a210a178a199a89a113a234,a89a15a121a210a89a199a48a27a123a128a154,a199a123a128a154
a204a7a52a183a243a207a1963a145a56a45a123Euc lid a1a91,Euclida1a91a33a80a52a126a116a98a41a37,a79
a143a116a98a117a243Euclida1a91a88a35,a52a139a63a66a123,a196a49a33a80a126a116a98a41a37,a0a143,a24a196
a123a135a73a1a91a123a86a127a126a116a77a193a88a123a30a127,a199a88a126a116a98a41a37,a28a7a46a47a123a27a6.
a7a152a128,a178a193a137a219a1a91,a28a88a126a116a98a41a37,a28a123a220a199a43a41a88a18a134,a4a150a149a199a18
a123a48a53a88a52a92a160a123a48a53,a28a86a65a65a23a210a127a119a10a245a234,a102a38a245a234,a88a52a88a44a21
a41,a44a49a41a116,a210a52a102a38a245a234,a139a63a135a123a24a71a47a60,a104a108a183a162a209a127a119,a43a41a248
a58a52a27a41a123a83a166a27,a106a166a27a136a11a16a21a25a123,a198a62a24a39a126a43a41a123a30a127a88a52a116a98
a41a37,a209a52a127a63a123a41a37,a198a126a127a63a123a41a37a252a117a255a174a89a171a1a91a123a175a11,a88a127a119
a52a253a160a11a9a56a234a199a116a98a41a37,a49a128a28a243a86a222a23a116a123a209a52a116a98a41a37,a196a49,a183
a123a41a37a52xe1e2e3,a894a199a209a52a37a222,a199a162a123a135a73a15a122a116a37a222a138a123a24a39a135a73,a196
a49a154a24a199a44a49a44a143e1,e2,e3a123a34a117a28a92,a132a149a183a162a52a243a24a1993a145a123a56a45,a0a143
a89a210a52a222a193a85a123a89a199a218a42(6.4)a90a5a193a123a218a42:
dei = ωijej,(6.5)
a89a30a127,a79a143a52a24a39a135a73a42,a196a49a89a171a34a117a28a92a52e1,e2,e3a123a34a117a28a92,a199a123
a248a106a52a24a39a135a73a42,a88a242a52a60a106a234,a49a3a152a42a52a60a106a123a155,a199a34a117a28a92a123
4
a248a106a52a60a106,a25a243,a248a106a52a24a39a135a73a42,a89a74a24a39a135a73a42a173a7a122a105,a79a143a199
a195a85a24a199a41a37a203a199a248a163a41a37a123a7a248,a199a248a163a41a37a196a24a154a154,a203a198a117a35a139a245
a232? a35a139a52a24a199a135a73,a210a52a183a162a123ωia203ωij,a89a1a199a135a73a42a139a59a92a123a7a248,a33
a7a155a123a52ωij,a28a23a199a105a102a38,i,ja441a1163,a98a52a79a143a41a37a52a116a98a123a92a160a37a222,
a196a49ωija233a149i,ja52a39a233a193a123,a79a36,a28a178ωija85a196a24a1993 × 3 a123a48a106a123a155,a89a199
a48a106a52a39a233a193a123,a199a123a233a110a34a123a195a163a209a520,a63a37a233a149a233a110a34a199a52a39a233a193a123,
a196a49a144a1393a199a93a116a7a255a174a123a24a39a135a73a42,a28a7a126a41a37a117a207a196a1a91a123a89a171a60
a89,a243a197a166a105a7a108,a197a166a89a24a199a212a14a243a0a14a35a196,a0a143a199a123a160a152a210a52a30a45a123
a60a106,a79a36,a89a41a37a52a30a45a123a60a106,a89a171a60a106a243a197a166a222a52a24a199a35a106a123a60a106.
a79a143a243a197a166a222,a243a196a197a166a222,a93a116a123a35a106a52a30a45,a144a139a24a199,a98a52a7a207a196a1
a91a123a155,a60a89a117a122a133a236,a44a21a89a199a41a37a52a203a245a149a24a199a35a106a139a7a248,a44a49a52a245
a35a106a123a60a106,a79a36a89a131a45a210a139a74a7a248,a89a7a248a210a52a28a70d(dei),a183a89a44,a28a126
a222da123a155,da126a220a39a520,a196a49a28a178a89a199a7a248a85a241a117a123a155,a210a122a116dωij a52a24a199a42
a6,a44a49a126a217a198a123ωa117a44a43,a89a42a6a52
dωij = ωik ∧ωkj,(6.6)
a28a122a116a89a248a6a24a28a48a199,a89a52a139a63a66a123,a79a143ωija52a24a39a135a73a42,a28a178a199a135
a73a123a155a522a39a135a73a42,a13a243a141a30a52a220a199a24a39a135a73a42a35a198,a196a49a15a522a39a135a73
a42,a79a36a89a28a48a199a88a175a216,a89a28a48a199a58a158a7a155,a199a162a83a44a228a196a107a114a199a123a117
a159,a89a28a48a199a23a248a133a236,a217a34a58a158a59a92,a79a143a89a74ωij a52a39a233a193a123,a196a49a152
a42i negationslash= j a123a155,a190a152,a152a42i = 1,j = 2,a0a143k = 3,a89a52a79a143k a7a52a127a1491,a149
a52a139ω11 = 0,a13a7a52ka127a1492,a0a143a139ω22 = 0,a196a49a89a28a48a199a42a23a248a133a236,a141
a30a144a139a24a49,a105a59a92a143,a28a164a44a49a122a116a24a199a1a47a60a111,a210a122a116
dω12 = ω13 ∧ω32 =?ω13 ∧ω23,(6.7)
a89a199a218a42a7a155a244a234,a183a162a243a89a199a60a111a210a141a116a24a199a99a123a60a89,a51a248a28a162a39a135a232
a73a123a30a127,a24a196a144a139a24a199a56a45,a76a150a24a196a88a52a178a193a210a523a145a56a45,a44a52a183a162a25
a243a139a220a199a56a45,a24a199a52a41a37a158a106a196a123a56a45,a523a145a123;a12a24a199a52a183a1622a145a123a77
a193,a196a49a183a139a24a1992a145a77a193a164a139a24a1993a145a123a56a45,a893a145a56a45a52a199a41a37,a79a36a152
5
a42a24a199a41a37,a28a82a199a198a154a123a155,a183a162a128a199a210a61a107a116a77a193a222a86a234,a89a248a6a210a139
a199a61a107,a25a243a199a139a199a214a35a119a41a14a145a45,a25a243a52a203a177a45a234,a14a145a52a203a177,a139a24
a178a203a177,a13a114a199a123a203a177a196a196a123a56a45a210a52a183a198a117a123a77a193,a183a162a119a198a117a123a77a193a143
a142a56a45,a252a51a24a199a198a154a123a196a139a92a160a35a37a222a210a196a14a145,a149a52a232a196a14a145a45,a199a210
a54a183a162a31a113a145a123,a139a24a30a24a30a123a34,a33a59a92a123a14a145a45a52a199a123a14a145a52a134a34,a0
a143a199a52a134a34a45,a183a65a248a178a199a7a48a196a24a178a77a6,a28a139a80a245a77a6,a154a24a202a77a6a52
a30a134a34,a0a143a139a80a245a77a6,a114a199a77a6a196a24a199a56a45,a89a210a52a183a162a123a14a145a45,a89a52
a134a34a123a60a89,a183a162a25a243a41a123a60a89a52a203a177a45,a89a199a10a39a52a135a232a73a176a62a24a199a99a123
a10a39,a210a52a128,a28a88a52a255a88a24a199a56a45,a13a52a28a243a255a88a220a199a56a45,a63a37a89a220a199a56
a45a131a45a139a183a35a123a7a248,a24a199a52a203a177a196a196a123a56a45,a24a199a52a183a162a123a142a56a45,a15a210
a52a198a117a123a77a193,a89a220a199a56a45a131a45a139a183a196a128a123a89a199a7a248,a89a199a7a248a139a63a135a244a234,
a173a7a244a234,a79a143a139a5a193a123a7a248
dω12 =?Kω1 ∧ω2,(6.8)
a141a30a52a77a193a222a123a42a6,a89a52a79a143Ka52Gaussa77a71,ω1 ∧ ω2a52a193a232a221a222,a196a49
a141a30a52a77a193a222a123a117a159,a38a30a52a24a199a192a220a123a135a73,ω12a52a243a14a145a45E a176a62a123a24
a39a135a73a42,a89a199a24a39a135a73a42a123a105a135a73a127a149a141a30a123a42a6,a89a199a121a210a128a210Gauss
a77a71a144a203Riemann a221a222a139a7,a79a143a7a52a139a234Riemanna221a222a210a139ω12,a0a143a183
a162a141a30a123a42a6a144a203Riemanna221a222a139a7,a89a52Gauss a104a35a105a122a63a123a24a199a136a42,
a203Guassa209a250a122a105a88a122a234a139a89a143a248a6a24a199a7a248,Gauss-Bonneta218a42a210a52a183
a162a7a70a141a30a89a199a42a6a123a232a73,a183a162a25a243a139a24a199a85a20a123a77a193,a199a52a189a53a123,
a7a70a141a30a123a232a73,a70a241a199a123a138a117,a0a143a104a30a183a15a139a24a171a70a216,a79a143a141a30a89
a199a42a6a17a108a52d a24a199a192a220a123a155,a243a24a199a85a20a77a193a222a123a232a73a97a148a520,a47a34
a222,a199a97a148a127a149ω12a215a248a89a199a77a193a123a30a142a123a232a73,a13a152a42a77a193a52a85a20a123,a199
a150a139a30a142,a196a49a97a148a520,a89a23a108a52a70a123,a143a31a143a199a88a127a1490? a183a162a175a108
a139dω12 =?Kω1 ∧ω2,a98a89a199a7a248a88a52a243a24a1992a145a56a45a222,a199a52a243Ea89a1993 a145
a56a45a222,a196a49a183a162a144a21a234a2433a145a56a45a188a126Stokes a189a174,a13a2433a145a56a45a123a155,a89
a199a77a193a2433a145a56a45a176a62a210a139a30a142a234,a28a7a178a89a199a77a193a12a1163a145a56a45a86,a14a143a12
a17? a210a52a154a154a7a201a24a199a252a89a154a41a198a154a123a35a37a222,a166a233a155a128,a89a210a52a196a162a123a37
6
a222a156.a196a49a89a199a77a193a137a7a139a199a37a222a156,a154a154a139a199a35a37a222,a13a89a199a35a37a222a52Ea176
a62a123a24a199a154,a210a178a89a199a77a193a12a116Ea176a62a86,a34a38a139a24a199a77a193,a52a88a52a24a189a139
a199a37a222a156? a89a88a59a92a234,a243a219a92a123a30a127,a104a108a105a59a92,a28a85a5a43a41,a177a34a85a74
a37a222,a210a139a234,a52a88a52a21a234a243a114a199a77a193a201a24a199a37a222a156,a89a52a1a91a176a62a196a162a114
a14a123a175a11,a202a47a95a192a210a180a89a199a175a11,a15a210a52a128,a219a92a23a108a44a49a85a37a222a123,a28
a139a219a92a43a41,a28a178a43a41a73a222a85a5a117,a104a108a210a139a199a37a222a156,a98a52a89a199a52a219a92a123,
a21a88a21a106a223a116a114a199a123a77a193,a88a24a189a44a21,a0a143,a183a222a39a46a178a89a44,a7a89a248a123
a155,a23a140a227a142a89a199a37a222a156a139a73a154(singularity),a7a48a128,a243a5a193a1a199a67a176a62a139
a1a199a37a222a156a123a190a6,( a67a68a63a210a161) a33a38a30a123a190a6,a199a123a73a154a210a243a198a154,a178a44
a89a199a198a154,a53a196a139a48a53a152a37a222,a248a234a198a154a131a105,a210a189a234a24a199a37a222a156,a98a52a198a154
a52a24a199a73a154,a199a52a196a139a121a22a241a117a123a241a21a154,a196a49a199a52a199a73a154,a145a19a199,a196a139a37
a222a209a53a198a154a18,a198a154a164a52a24a199a73a154,a198a154a210a35a196a24a199a188a5a86a123a24a199a154,a93a169
a119sink,a13a38a30a123a119source,a104a108a15a139a54a33a141a30a123a190a6,a44a89a74a190a6a44a49a23a241
a37a222a156a243a73a154a139a88a51a117a159,a152a91a195a85a199a123a88a51a123a117a159,a210a139a24a199a119a41a37a222a156
a123a141a41(index),a28a243a89a24a154,a34a38a139a199a241a193a123a73a154,a0a143a140a248a89a199a73a154a41a199
a66a203a87,a79a143a52a241a193a123a73a154,a196a49a243a66a203a87a222a123a154a123a37a222a52a113a92a104a189a123,a0
a143a25a243,a243a66a203a87a123a154a119a248a73a154a221a245a232a87a17? a152a42a221a24a87,a63a37a52a243a116a123a48
a53a221a24a87,a199a123a141a41a521,a152a42a53a139a123a48a53a210a52?1,a0a143,a243a183a123a222a193a190a6
a165,a195a88sink a164a52source,a141a41a209a521,a118a77a34a123a25a54a141a41a143?1,a73a154a105a133
a236,a79a36a141a41a44a49a82a127a91a138,a34a38a183a178a77a193a12a116a14a145a45a176a62,a12a116a203a177a45a176
a62,a63a37a227a142a139a73a154,a0a143a89a199a222a86a123a77a193a210a139a30a142,a89a199a30a142a210a35a104a149a89
a74a73a154,a196a49a202a226a218a42dω12 =?Kω1 ∧ ω2,a183a162a7a149Gaussa77a71a123a232a73a210a127
a149a73a154a123a141a41a90,a196a49a183a162a121a210a24a199a173a7a117a159,a88a88a28a82a127a91a24a199a37a222a156,
a34a38a199a144a139a139a33a199a73a154,a183a162a89a199a232a73a52a141a41a90,a253a52a178a154a199a73a154a123a141a41a28
a229a117a210a127a149a141a41a90,a89a176a105a7a155,a79a143a89a199a232a73a52a203a37a222a156a160a10a195a7a123,a196
a49a89a121a210a234a24a199a77a193a34a38a139a24a199a139a139a33a199a73a154a123a37a222a156,a243a73a154a123a141a41a90
a37a222a156a123a160a10a195a7,a199a127a149a0a199a232a73,a13a0a199a232a73a176a52a150a139a37a222a156,a196a49a210
a122a116a89a248a24a199a136a42,a183a242a128a24a40,a25a243a139a24a199a85a20a123a77a193,a82a24a199a37a222a156,a139
a139a33a199a73a154,a199a123a141a41a90a52a166a37a222a156a123a160a10a195a7a123,a89a52a79a143a199a127a149a141a30a123
7
a232a73,a13a141a30a232a73a202a253a150a139a37a222a156,a196a49a166a37a222a156a123a160a10a195a7,a143a31a143a89a199
a106a248a127a149Euler a43a117a106a17? a25a243a17a108a199a203a37a222a156a160a10a195a7,a28a210a127a82a24a199a37
a222a156,a7a48a128,a34a38a139a199a77a193,a28a178a77a193a73a191a234,a73a191a196a66a76,a154a199a66a76a52a174
a110a111,a233a149a89a174a110a111,a154a199a30a82a199a123a165a154,a174a110a111a82a199a123a173a101,a28a210a44a49a189
a24a199a37a222a156,a210a54a183a196a152a123,a44a186a154a241a86,a108a128a116a174a110a111a123a173a101a210a159a86,a233a149
a89a248a6a189a123a37a222a156a105a142a52a23a241a117,a166a28a243a30a222a123a89a171a154a123a141a41a127a149?1,a149
a52a199a243a186a154a123a141a41a521,a243a174a110a111a173a101a123a141a41a209a521,a98a52a243a30a222a154a199a154a141a41
a143?1,a196a49a178a89a141a41a28a229a117a123a155,a210a127a149a186a154a123a199a106+a193a123a199a106?a30a123a199a106,
a79a36a210a52Euler a43a117a106,a89a248a6a121a210a234Gauss-Bonnet a218a42.
4 Gauss-Bonneta218a42a123a77a18a249a97a126
Gauss-Bonneta218a42a93a116a139a126a123a30a127a52a77a193a139a30a142,a243a77a193a139a30a142a123a30a127,
Gauss-Bonnet a218a42a52a186a154+a186a154a123a105a110+a30a123a127a143a77a71(geodesic curva-
ture),a242a28a222a243a193a123Gaussa77a71,a5a193a52a24a196a123Gauss-Bonneta218a42
summationdisplay(pi?α) (a154) +summationdisplayintegraldisplay k
g(s)ds (a30) +
summationdisplayintegraldisplay integraldisplay KdA (a193) = 2piχ(M) (6.9)
a233a149a139a30a142a123a77a193,a62a24a92a73a52a30a142a186a154a123a154a77a71,a217a39a52a30a142a123a30a123a34
a77a71,a108a128a114a199a123a89a199a192a220a123a193a77a71,a196a49a28a139a24a199a139a30a142a123a77a193,a28a210
a82a30a142a123a154a77a71+a30a142a123a34a77a71+a193a77a71,a52Eulera43a117a106,a121a210a52a24a248a123.
a93a116Gauss-Bonneta218a42a33a139a126a123a52a139a30a142a123a60a89,a7a48a128,a243a24a199Euclida178
a193,a34a38a139a24a199a174a110a111,a89a199a174a110a111a132a134a34a196a196,a132a149a56a45a52Euclida56
a45,Gaussa77a71=0; a34a38a30a209a52a134a34,a196a49a127a143a77a71a15a520,a79a36a89a199a210a52
a128summationtext(pi?α)a127a1492pi,a89a52a79a143a7a52a174a110a111,Eulera43a117a106a521,a141a30a7a127a1492pi,
a196a49a89a210a128a210a174a110a111a174a110a131a90a243Euclida178a193a222a127a149pi,Gauss-Bonneta218a42a52
a174a110a111a174a110a90a218a42a243a24a196a60a111a123a77a18,a89a199a10a39a173a7a244a234,a199a210a52a114a199a14
a145a45a123a10a39,a183a128,a132a89a199a14a145a45,Maxwella48a199a210a52a89a199a60a89a123a77a18,a28a116
a212a174a222a97a126a123a30a127,a28a123a56a45a524a145,a523a145a56a45+1a145a30a45,a524a145a123a98a85a251a22
a111,a0a143a7a44Maxwella48a199a123a155,a28a7a126a24a199a203a177a45,a34a19a222a52a24a199a133a123a134
8
a34a45,a199a139a199a77a71,a183a162a123a77a71a52Gaussa77a71a198a49a193a232a195a163,a13a89a199a77a71a52
a1992a39a135a73a42,a178a44a43a89a199a77a71a52a85a20a123a30a71a85a241a117a210a52Maxwe lla48a199,a196
a49,Maxwella48a199a123a1a91a242a181a52a58a158a59a92a123,a210a52a79a143a45a142a524a145a123a56a45,a196
a49a52a442a145a56a45a106a223a1164a145,a0a143a89a199a77a71a79a143a52a24a1992a39a135a73a42,a164a52a39a233
a193a123,a79a36a2434a145a56a45a176a52a24a1994×4a48a106:
(Fαβ =
0 E1 E2 E3
E1 0?B0 B2
E2 B0 0?B1
E3?B2 B1 0
,(6.10)
a89a199a48a106a176a62E1,E2,E3 a52Electric Potential,B0,B1,B2a52Magnetic Po ten-
tial,a15a210a52a158a51a203a30a51,a89a74a209a52a48a106a176a62a123a60a106,a44a43a132a89a199a48a106a196a44a43
a1232a39a135a73a42a52a85a20a123,a253da89a199a42a6a123a135a73a1430,a210a52Maxwell a48a199
d(Fαβdxα ∧dxβ) = 0,(6.11)
Gauss-Bonneta218a42a52a24a199a187a36a222a105a245a220a35a123,a93a116a183a85a123a218a42a17a88a132
a149Gauss,a15a88a132a149Bonnet,Gaussa144a41a234a174a110a111a123a60a89,a132a127a143a174a110a111a41a116
a174a110a111,Bonneta15a150a139a41a95a192a123a97a126,Bonneta178a174a110a111a77a18a116a127a63a77a34,a198
a178a127a63a77a34a123a127a143a77a71a232a73a44a143Gauss-B onneta218a42a123a232a73,a104a35Bonneta52
a27a41a33a7a155a123a1a91a166a27,a198a243a135a73a1a91a41a234a58a158a228a253a123a224a26,a183a88a11a28a162
a234a137a245a232,a183a230a130a28a162a234a137a89a24a92a73a123a106a166a58a158a7a155,Maxwella48a199a52a199a123
a1a47a60a89,a89a199a58a158a139a126a255,a183a139a24a159a169a57a243a152a35a123a21a41a166a22a222a21a44,a11a248
a119a21Gauss- Bonneta218a42a166Maxwella48a199a22,a183a89a123a142a245a7a154a243a89a159a169a57a176a62
a139.
9
