1、a145a24a89 a135a73a166a232a732001a3510a21911a1341 a135a232a73a123a229a205: a58a238a166a116a89a26a251a89a116a135a232a73,a33a7a155a123a220a199a124a52a58a238(Issac Newton, 1642-1727)a203a116a89a26a251(Gottpied Leibniz, 1646-1716), a135a232a73a210a52a198a162a21a25a123. a7a149a58a238, a139a108a85a123a52a
2、198a41a89a199a211a42a52a243a166a9a123a30a127, a15a142a7a28a162a123a181a106a164a7a66, a0a199a30a127,a15a210a5217a45a22a0a199a30a127, a78a179a166a60a105a184a51, a78a179a139a234a105a245a124. a198a243a93a41a74a24a76a166, a79a143a166a60a123a7a248, a166a68a56a34a234, a198a210a195a27a243a27a176a41a7a149a13
3、5a232a73a123a89a74a211a42. a116a89a26a251a52a24a199a200a48a193a209a58a158a128a132a123a124, a106a166a52a198a123a108a85a123a24a92a73,a198a123a108a85a116a13a115a1a27a70a200a48a193a209a139. a198a162a220a124a131a45a139a154a112a88, a52a79a143a112a88a120a52a135a232a73a123a21a25a86. a89a199a112a88a52a88a115
4、a123, a15a150a139a31a143a63a66. a34a159a222a52a116a89a26a251a62a24a199a21a44a7a149a135a232a73a88a169a123a124, a198a123a88a169a2431684a35a21a44. a58a238a41a89a199a211a42a0a149a116a89a26a251, a13a116a89a26a251a21a44a88a169a0a149a58a238, a58a238a139a234a89a199a211a42a128a150a139a21a44a31a143a127a91a123
5、a192a220. a13a116a89a26a251a88a98a21a44a234a89a74a192a220, a51a30a164a90a126a234a24a74a110a82, a15a142a183a162a25a243a164a243a126. a0a143a128a117a220a199a124a139a24a199a112a88, a76a150a209a52a203a106a166a150a139a7a248a123a124a243a0a176a5a196a123a60a89, a89a88a52a24a199a31a143a139a63a135a123a47a60.2
6、a135a232a73a228a253a189a174a135a232a73a52a106a166a176a62a105a173a7a123a48a193. a150a149a31a143a52a135a232a73a17a218a183a46a135a73a123a21a25a203a136a11a14a21a25a43a41a58a158a139a7a248, a79a143a136a11a14a21a25a43a41a131a128, a106a166a204a7a123a248a123a210a52a207a196a60a106, a207a196a220a28a106a123a7a2
7、48, a139a171a171a123a7a248. a183a162a127a119, a60a106a139a171a171, a139a34a117a123, a58a34a117a123, a174a110a60a106a127a171a171a60a106,a0a143a7a14a143a248a143a207a196a60a106a123a117a159? a183a162a209a127a119, a60a106a44a49a126a77a34a117a44a43, a152y = f(x) a89a30a77a34. a243a89a30a77a34a123a154a154,
8、 a152a42a199a52a44a49a135a73a123a155, a0a143a199a243a154a154a139a199a35a34. a135a73a210a52a178a89a199a77a34a126a199a123a35a34a117a207a196a199a123a117a159. a196a49a15a127a149a128a199a52a178a60a106a34a117a154, a341a117a154a131a128, a44a49a28a1a62a1a198a1a248, a44a49a15a174, a79a36a44a49a122a116a106a24
9、1a117. a106a166a7a52a21a234a122a116a106a241a117, a15a52a105a7a155a123. a196a49a135a73a76a150a52a128a126a77a34a123a35a34a117a207a196a77a34a123a117a159. a232a73a117a122a0a234,a79a143a232a73a34a19a222a76a151a89a229a117, a199a52a7a15a174a193a232. a0a143a34a38a178a193a222a139a24a199a75a173, a132a77a34a11
10、7a41a143a30a142, a199a123a193a232a139a245a76, a203a177a123a193a232a139a245a76, a89a176a123a175a11a52a232a73a123a13a41, a15a52a232a73a173a7a123a248a123. a79a36, a34a19a222, a232a73a123a21a48a243a135a73a131a3. a232a73a104a30a15a150a139a24a189a123a189a66, a232a73a210a52a139a199a244a33a123a10a39. a77a34
11、a196a140a194a123a75a173a24a196a46a27a6a126a134a34a117a5a163, a38a122a5a163a123a77a34a74a149a28a123a30a142a123a30a127, a210a139a199a244a33, a210a52a89a199a75a173a123a193a232. a196a49, a15a13a211a131, a232a73a123a21a48a243a135a73a131a3. a165a45a89a220a199a175a11a80a54a150a139a7a248, a98a52a217a34a89a7
12、a248a58a158a123a183a35. a232a73a139a88a245a52a135a73a123a39a228a174. a7a48a128, a34a38a28a70a89a30a134a34a203a220a30a18a34a196a196a75a173a123a193a232, a89a220a30a18a34, a24a199a52s = a,a24a199a52s = x, a28a7a86a174a89a199a75a173a123a193a232, a52a199a189a232a73integraltextxa f(x)dx, (a214a42f(x)a189a
13、232a73a44a x). a89a52a104a35a116a89a26a251a123a110a82, a89a199a232a73a123a110a82a16a196a89a248, a79a143a232a73a15a52a83a44a24a199a90, integraltexta83a44a90(sum). a34a247a193a232a24a30a132s = a a123a134a34a42a30a142, a12a24a30a52a127a63a123x, a28a178xa89a30a134a34a35a196a123a155, a210a122a116a24a199x
14、a123a60a106, a89a199a60a106, a183a119a199A(x),a210a52a183a67a222a123a193a232, a52a199a232a73, a196a49a199a52a24a199a106a248, a166xa139a7, a196a49a52xa123a60a106. a89a199a60a106a203a77a34a48a199y = f(x)a89a199a60a106a139a183a35a7a248. a143a31a143a139a183a35a123a7a248a17a218a105a59a92a143a23a23, a34a1
15、52a70A(x) = integraltextxa f(x)dxa123a135a73, a70a199a123a135a73a108, a210a52a128, a70s = x,x + x a196a140a196a123a89a199a66a30a75a173a123a193a232. a25a243a152a42a28a252xa248a123a155,a183a46a105a142a52a23a241a117a234, a89a199a244a33a210a52f(x). a196a49a105a142a52a23a241a117A(x)a89a199a60a106a123a135
16、a73a210a52f(x), a79a36dA(x)dx = f(x). (1.1)a89a210a52a135a73a51a232a73a123a228a253a123a7a248. a89a199a7a248a128A(x)a52a24a199a232a73, a70a199a123a135a73a123a30a127, a210a122f(x). a89a199a24a196a143, a119a41a135a232a73a123a228a253a189a174. a183a44a3a243a8a13a39a135a232a73a123a30a127, a41a170a88a195a1
17、43a31a143a89a52a24a199a135a232a73a123a228a253a189a174, a79a143a24a196a143a178a89a199a7a248a42a85a196 integraldisplayf(x)dx|ba =integraldisplay baf(x)dx (1.2)a111a231. a38a30a232a73a52a199a88a189a232a73(indefinite integral), a88a189a232a73a52a199a60a106, a38a422a52a60a106a243ba123a138a62a86a60a106a24
18、3aa123a138, a127a149a89a199a189a232a73(definite integral). a196a49a44a89a199a7a248a127a119a7a70a232a73a123a155, a144a137a7a70a24a199a60a106, a199a123a135a73a52a46a127a123,a210a52f(x), a253a135a73a52a46a127a123. a196a49a89a248a135a73a203a232a73a203a229a117a234. a144a35a123, a232a73a127a149a135a73a123
19、a39a228a174, a139a234f(x), a7a73a24a199a60a106, a199a123a135a73a127a149f(x), a52a199a39a228a174.a79a36a135a1a232a73a139a183a35a123a7a248.3 a245a195a135a232a73a222a193a89a123a52a24a199a35a106a123a135a232a73. a5a193a89a176a145a123, a7a245a35a106a123. a245a35a106a123a155,a139a99a123a25a54, a52a31a143a2
20、48a123a17a218a183a46a233a149a245a35a106a123, a183a162a11a88a23a47a123, a11a23a220a199a35a106a123a60a111, xa203y, a0a143a183a162a127a119a89a199a30a127a135a73a123a10a39a123a77a18a52a160a135a73,a127a149xa203ya73a13a70a135a73. a232a73a123a10a39a77a18a52a173a232a73. a19a173a232a73(double integral)a52a243
21、2a145a123a60a111, a243a176a145a123a60a111a52a245a173a123. a11a232a145, 2a145a123a60a111a210a139a234a75a173,a183a162a119a199, a0a143a199a123a30a142a119a199. a196a49a232a73a123a24a199a7a108a77a18a52a24a1992a173a232a73,a202a47a232a73a178xa73a196a66a227, a108a128a82a66a227a242a198a222a89a199a60a106, a70
22、a24a199a90. a2432a173a232a73a123a30a127, a48a27a15a52a178a75a173a73a196a66a76, a108a128a82a154a24a66a76a123a193a232, a243a217a222a60a106a138a198a222a199a123a193a232, a108a128a70a199a123a90. a105a88a122a234a123, a34a38a60a106a80a123a155, a195a88a28a152a91a87a28a123a75a173,a244a33a52a24a248a123, a196a
23、49a89a244a33a210a522a173a232a73I =integraldisplay integraldisplayf(x,y)dxdy. (1.3)a2432a145a123a30a127, a4a150a176a145a123a30a127, a24a199a173a7a123a25a54a52, a183a162a25a243a1392a199a35a106x,y,a166a35a106a14a143a248a218a196a49a183a25a243a166a35a106, a166a35a106a104a108a52a243a135a232a73a176a52a105a
24、173a7a123a24a199a205a27, a79a143a105a245a123a175a11a52a23a28a123a35a106a52a100a160a10a122a55a104, a139a30a166a35a106, a175a11a210a193a47a59a92a154a234, a210a44a49a137a251a234. a25a243a183a166a35a106a213braceleftBiggx = x(xprime,yprime)y = y(xprime,yprime) (1.4)a217a165, (xprime,yprime)a52a12a105a24a
25、28a43a41. a183a162a21a25a24a199a47a34,a243a176a145a123a30a127,a135a73a123a198a27, a183a162a85a196dx dy, a89a52a24a199a198a27, a14a143a198a17a218dx dya243a135a232a73a222a52a33a135a201a123a10a154. a31a143a119a135a73a218a31a143a52dxa218a89a199a52a104a118a234a106a166a27a1a186a35a123a47. a14a1433a248a189
26、a135a73a123a189a66a203a196a190a31a143a52dx, a89a199a105a102a38, a44a49a41a116a105a119a63, a88a44a178a199a89a56a249a137a7a139a24a189a123a30a45. a196a49a183a106a106a141a141a128a139a24a199dx. a243dx,dya89a171a135a73a131a45a7a79a193a198a27. a31a143a119dx dya218a89a199a175a11a205a133a236a234, a28a152a42d
27、x,dya253a252a52a31a143a209a88a56a249, a198a234a49a128a52a31a143a192a220a205a52a24a199a105a135a201a104a10a123a175a11. a243a89a48a193a139a24a199a76a123a159a90, a210a52a90a159a105a83a106a90a105a135a73. a34a189dx dya89a199a198a27a52a39a233a193,dxdy = dy dx. (1.5)a89a199a175a11a210a56a249a59a92a234. a79a
28、143a198a27a152a42a52a39a233a193a123a155,a104a108dx dx = 0. a47a34a222, a79a143dx dx = dx dx, a196a49dx dx = 0, a243a39a233a193a123a198a27a131a5,a178dx dya23a196a35a106, a79a143a198a27a52a39a233a193a123, dx2 = 0, a196a49a210a150a139a176a39a123a192a220a234. a89a248a122a116a123a83a106a119a41a105a83a106
29、. a89a199a83a106a105a201a123. a139a24a199a193a47a123a136a88a213a166a35a106a218a42a143dxdy = (x,y)(xprime,yprime)dxprime dyprime. (1.6)a34a38a183a162a123a135a73a126a123a52a160a135a73, a196a49dx = xxprime dxprime + xyprime dyprime,dy = yxprime dxprime + yyprime dyprime. (1.7)a25a243a126a105a198a27a24a
30、198, dxprime dxprime = dyprime dyprime = 0. a13dxprime dyprimea79a143a198a27a52a39a233a193a123, a196a49a52a166a80a198a49x = x(xprime,yprime),y = y(xprime,yprime) a123a196a44a7 (x,y)(xprime,yprime), a89a199a110a82a52a196a44a7, a52a143a199a160a135a73a196a196a123a113a239a42, a196a49dxdy = (x,y)(xprime,
31、yprime)dxprime dyprime. (1.8)a89a199a166a28a52a183a162a173a232a73a166a35a106a123a24a199a7a248.a183a162a127a119a173a232a73a7a52a166a35a106a123a155, a199a97a148a198a222a196a44a7. a196a49a89a199a136a88a210a52, a233a173a232a73a123Integral, a253a232a73a5a123a42a6, a178a232a73a82a191a171, Integrala52a24a1
32、99a135a73a245a49a42, a198a27a52a39a233a193a123. a196a49a34a38a245a173a232a73a1393a145, 4a145a116n a145a123a56a45, a245a173a232a73a123Integrala44a23a196a52a105a83a106a123a245a49a42, a0a143a166a35a106a210a7a108a233a234. a89a176a62a139a24a154a135a201a123a143a48, a79a143a47a158, a28a7a121a210a166a35a106
33、a123a218a42a123a30a127, a34a189a196a44a7a52a116a123, a88a108a123a155, a198a222a196a44a7a123a253a233a138, a38a199a52a116a123. a89a199a52a176a145a1a91a135a201a123a192a220, a210a52a56a45a139a1994a53(Orietation),a28a221a123a30a127, a1392a199a35a39a221a123a48a53. a221a123a30a127, a34a38a149a234a48a53a123
34、a155, a196a44a7a52a139a138, a79a36a183a162a24a199a136a88a52a245a173a232a73a123Integrala97a148a52a24a199a105a83a106a245a49a42, a52dx,dya123a245a49a42, a198a27a52a39a233a193, a89a248a166a35a106a113a92a44a49a233a123,a104a108a183a144a41a2342a145a123a190a6. a176a145a52a105a210a23a123, a51a248a123.a105a19
35、8a27a52a201a122a105a254, a52a88a204a139a176a39a123, a196a49a7a118a59a92, a178a48a24a5,a210a520.4 a105a135a73a222a193a89a234a89a143a248a24a171a7a248, a4a150a89a7a248a164a205a7a80, a183a162a89a176a127a135a232a73a123a30a127,a24a199a173a7a123a189a174a52a194a245a189a174(Greens Theorem). a210a52a128, a34a
36、38a28a139a199a75a173,a243a30a142a222a123a135a73a52a44a49a35a143a75a173a222a123a135a73, a52a24a199a24a173a232a73a90a19a173a232a73a123a7a248, a89a52a199a58a158a173a7a123a7a248. a7a48a215a12a115a71a139a24a253a66a86, a89a116a89a199a7a248, a198a128a143a89a52a114a199a135a232a73a123a228a253a189a174, a183a5
37、2a51a63a123. a89a248a123a7a248a25a243a47a158a85a194a245a189a174a123a30a127, a128a128a52a85a196a139a232a73,integraldisplayAdx + bdy = (Bx Ay )dxdy. (1.9)a152a42a139a24a199a175a11, a139a30a127a28a44a49a144a11Integral, a88a7a11a217a199, a0a143Integrala210a52a178a24a199a24a39a135a73a42a35a143a220a39a135
38、a73a42, a89a14a143a35a17a218a218a42a189a174a52a89a248a6: a183a210a90a156a24a199a105a135a73, a183a162a166a98a89dx dy a52a24a199a245a49a42, a52a24a199a105a83a106a123a24a199a42a6, a210a54a183a162a202a47a245a49a42a24a248,a88a98a152a36, a233a149a89a248a123a42a6, a183a162a164a44a49a189a66a199a24a199a135a7
39、3,d(Adx + Bdy) = dAdx + dB dy = Aydy dx + Bxdxdy. (1.10)a119a105a135a73(Exterior differential calculus). a105a135a73a105a59a92, a34a247a139Adx + Bdy,a199a123a135a73a210a52a135a73a199a123a248a106, a15a210a52a135a73a60a106. Aa166Ba52x,ya123a60a106, a196a49a210a135a73A,B . Aa123a135a73a210a52Axdx + Ayd
40、y, Ba123a135a73a210a52Bxdx + Bydy, a44a52Axdxdx = 0 a210a122a116Aydy dx, a145a19a49a210a122Bxdxdy. a98a52a79a143a198a27a52a39a233a193a123, a196a49a210a122(Bx Ay), a89a52a194a245a189a174a176a622a173a232a73a123a248a106, a196a49a194a245a189a174a178a92a39a232a73a35a196a220a39a232a73, a199a123Integrala34
41、a19a222a52a199a105a135a73. a44a49a23a241a1055a135a73a52a105a201a123a192a220, a79a36a28a44a49a178a232a73a82a191a171, a210a128a183a162a252dx,dya5a24a199a105a83a106, a233a89a199a105a83a106a139a199a105a135a73, a105a135a73a105a59a92, a210a52a34a38a135a73a200a49a123a30a127,a217a34a52a233a154a49a248a106a13
42、5a73, a136a42a183a122a116a24a199a245a49a42, a89a199a245a49a42a123a39a106a176a24a199. a42a143a60a106a210a35a143a24a39a135a73a42a234, a196a49a39a106a176a24a199, a79a36a210a42a143a198a117a52ka39a123a155,a122a116a24a199k + 1a39a123a135a73a42. a89a199a52a194a245a189a174a165a152a91a178a77a34a135a73a123a13
43、5a73a42a35a143a75a173a135a73a42, a24a173a135a73a35a143a19a173a135a73a123a218a42. a89a199a210a105a80a234, a79a143a89a176a193a139a24a199a105a83a106, a196a49a178a89a199a135a73a42a198a229a117, a126a24a199a105a198a27, a135a73a123a198a27a52a39a233a193. a108a128a17, a25a243a183a139a24a199a135a73, a199a178k
44、a39a123a105a135a73a42a35a143k + 1a39a123a105a135a73a42, a89a248a6a210a178a89a199a105a135a73a42a165a45a201a234a24a199a99a123a136a232, a44a49a135a73, a89a199a135a73a203a202a47a123a135a73a88a24a248, a199a52a178ka39a35a143k + 1a39, a135a73a24a196a143a15a52a28a24a39. a89a199a105a135a73a52a33a0a30a127Frob
45、enius, Daubouxa90a183a123a144a19Elie Cartana90a159a117a123. a198a162a33a240a90a159a89a199a10a39a52a233a149a24a39a135a73a42, a52Frobenius, Daubouxa90a156a123a24a39a135a73a42. a13Elie C artana52a27a41a123a115a71, a52a183a123a144a19, a198a57a89a52a19a27a45a22, a15a210a52a222a199a45a22a33a149a76a123a1a9
46、1a166a27, a27a41a174a169a76a166a123a115a71. a183a46a89a171a115a71a105a52a220a41, a198a88a41a47a123a217a196, a219a41a106a166, a30a158a213a49a52a113a92a99a123. a139a24a35, a198a201a234a24a160a49, a52a21a137a219a197a166a22(Analytical Mechanics), a198a178a105a135a73a123a10a39a44Frobenius, Daubouxa44a24a
47、39a42a123a189a66a77a18a116a176a39a42, a196a49a114a199a123a105a135a73a52Elie Cartana90a159a117a123, a89a52a139a126a123a192a220. a89a199a105a135a73a139a219a5a123a25a54: a52a126a220a39a131a128a127a1490.d2 = 0, (1.11)a253a89a199a105a135a73a126a220a39a127a1490. a183a162a7a121a210(1.11), a210a52a233a195a8
48、8a24a199ka39a135a73a42,a135a73a24a39a210a35a143k + 1a39, a220a39a210a35a143k + 2a39a135a73a42, a199a24a189a520. a7a121a210a89a24a154, a183a121a210a233a149a60a106a233a234, a210a113a234. a196a49a183a7a121a210a233a149a127a63a123a60a106f, a178a89a199d, a105a135a73a126a220a39, a210a127a1490, a253d2f = 0
49、a210a113a234. a0a143a143a31a143a17a218a79a143a23a108a183a7a121a210d2 = 0, a144a7a121a210d2a42a126a243a144a139a24a49a222a233a210a113a234, a89a52a79a143a199a52a34a117a123, a196a49a152a42a34a117a24a49a139a89a199a117a159, a0a143a114a199a123a90a210a127a1490. a0a143a24a49a123a155, a209a52a24a199a60a106a198a222a24a28dx, a183a25a243a160dxi, a210a52a34a189a243a176a145,a243na145,xia210a52x1a116xn, a243a176a145a30, a152a42a139a24a199a60a106f,fa52x1,xna123a24a199a60a106, a233a149a89a199a60a106, a126a105a135a73a220a39, a24a189a127a1
