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
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
