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
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
