1、2000 (1) 1 2 0 2x xdx . . 4 11 2 22 2 00 0 2 1 1 1 sin cos 4 x x dx x dxx t tdt 2 222 2 3 21 xyz 1, 2 , 2 . 122 146 xyz . 222 , 2 3 2 1 Fxyz x y z 1, 2,2 1, 2,2 1, 2,2 1, 2, 2 2 2 , 1, 2 , 2 4 8, 1, 2,2 6 12. | | | x y z Fx Fy Fz 122 146 xyz 3 30 xy y . 2 1 2 C yC x . p y 3 0, pp x 3 . p Cx 32 2 112
2、 2 , 22 C CC yC xd xCxC C x 4 1 2 3 12 1 1 23 2 3 120 x ax ax a . - 1 . 12 1 1 1 2 1 1 12 1 1 23 2 3 0 1 1 01 1 12002 310 0313 aaa aaa a a 1 a 2 3 . 3 a 2 . (5) A B 1 , 9 A B B A PA . 2 . 3 1 , 9 PA B PA B PA B A B A B A B PA B PA B PAPB PAPB 11, PA PB PAPB PA PB 2 1 1, 9 PA B PAPB PA PA 2 . 3 1 , f
3、 xgx 0 fxgxfxgx axb A f xgb fbgx B f xga fagx C f xgx fbgb D f xgx faga A . 2 0, fx fxgx fxgx gx gx axb fxfb gx gb f xgb fbgx A . 2 2222 1 :0 , Sx y z az S S A 1 4 SS xdS xdS B 1 4 SS ydS xdS C 1 4 SS zdS xdS D 1 4 SS xyzdS xyzdS C . S 0 x 0 y A B D C . , 11 44 SSS zdS zdS xdS 3 1 n n u A 1 1. n n n
4、 u n B 2 1 n n u C 21 2 1 . nn n uu ( D ) 1 1 . nn n uu D . D A B C . 2 1 1 ln n n n 22 1 1 ln n n nn u nnn A 1 1 1 n n n 2 11 1 n nn u n B 1 1 1 1 n n n 21 2 111 111 212 nn nnn uu nnn c . 4 n 1 , m mn n 1 , m A 1 , m 1 , m . B 1 , m 1 , m . C 1 , m 1 , m . D 1 , m A 1 , m B . D . . A 1 , m 1 , m 1
5、, m 1 , m 1 , , m rm 1 , m . 1 m 11 1, 0 , 0, 1 TT 1 1 . B . 1 m 11 1, 0 , 0, 1 TT 1 1 11 1, 0 , 0 , 0 TT 1 1 1 . C 1 , m 1 , m 1 , m 11 , , , mm rrm 1 , m 1 m 11 1, 0 , 0, 1 TT 1 1 . E D . 1 , m A 1 , m B rA rB 11 , , , mm rrm 1 , m . 5 , X Y X Y X Y (A) . EXE Y (B) 22 22 . EX EX EY EY (C) 22 . EXE
6、 Y (D) 22 22 . EX EX EY EY B . , , , , ,Cov Cov X Y X Y Cov X X Cov X Y Cov Y X Cov Y Y Cov X X Cov Y Y DX DY 22 22 ,0 0Cov D X D Y EX EX EY EY B . 1 0 2s i n lim . 4 1 x x ex x e x 11 00 11 00 2s i n 2s i n2 lim lim 1 1, 44 1 11 2s i n 2s i n lim lim 0 1 1, 44 11 xx xx xx xx ex ex xx ee xx ex ex xx
7、 ee xx , x x zfx y g yy f g 2 . z x y 12 2 1 zy yf f g xyx 2 11 11 222 12 2 22 223 121 12 2 232 3 11 1 11 zxxy f yx f f f x f f g g xy y y y y x x xy ffx y ffgg yyx x 22 , 4 L xdy ydx I x y L 1, 0 R 1 R . 22 22 , 44 yx PQ xyxy 22 2 22 4 , 0 , 0 4 PyxQ xy xy xy cos : 2 sin x t C yt 0, 2 , tC 22 0. 4
8、LC xdy ydx xy 2 2 22 22 2 0 1 2 44 LC xdy ydx xdy ydx IId t xy xy 0 x , S 2 0, x S xf x dydz xyf x dzdx e zdxdy f x 0, 0 lim 1, x fx f x . 2 2 0, x S x xfxd y d zx y fxd z d xez d x d y xfxfxx fxed V S S 2 0, 0 x xf x f x xf x e x 2 11 1, 0 x fx fx ex xx x x e fxeC x 2 00 lim lim 1, xx xx eC e fx x
9、2 0 lim 0, xx x eC e 10 C 1 C 1. x x e fx e x 1 1 32 n n n n x n . 1 11 1 2 1 32 3 1 lim lim lim 3 321 2 31 1 3 n n n n nn n nn n n n n a a n n 3 R 3, 3 3 x 3 11 2 32 n n n nn 1 1 n n 3 x 3 x 32 11 1 1 32 32 nn n nn nn nn n 1 1 n n n 1 2 1 32 n n n n n . 3 x . R 0 P 0 P 0 k . 0 P . 1 , O 0 OP x 0 P
10、,0,0 R 2222 x yzR , xyz 0, 0, yz 2 22 2 22 xkxRyzd V x kxRyzd V 2 22 223 2 22 5 22 000 5 4 8s i n 3 32 15 R xR yzd V x yzd V R d V ddrr d r R R 2 22 2 223 6 2 28 31 5 x xR yzd V Rxd V R xyzd V R 4 R x . ,0,0 . 4 R 2 O 0 P 0 P O z 222 2 x yzR z , xyz 0, 0, xy 223 223 kz x y z dV z kx y zd V 2c o s 22
11、2 4 22 000 5 4s i n 3215 R x yzd V dd r d r R 2c o s 222 5 22 000 67 2 0 6 4s i n c o s 64cos sin 3 83 R zx y zd V d d r d r Rd R 5 . 4 zR 5 0, 0, 4 R . f x 0, 00 0, cos 0, fxd x fx x d x 0, 12 , 12 0 ff . 0 , Fx ftd t 00 , FF 00 0 0 0 0c o sc o sc o s s i ns i n | fxx d x x d F x Fx x Fx x d x Fx x
12、 d x 0 sin Gx Fx t d t 00 , GG 0, sin 0, F 0, . 00 FFF Fx 0, , 1 0, 2 , 12 0 FF 12 0 ff 6 A * 1000 0100 , 1010 0308 A 11 3, ABA BA E E 4 . B A * A 1 . A 1 * , AA A A A E 1 * n AA 3 * 8 A A 2 A 11 3, ABA BA E A * A * 23, B AB AA AB * 26 , E ABE 1 1 * 1000 6000 0100 0600 62 6 1010 6060 0306 0301 BE A
13、2 2 A 1 * , AA A A A E 11 * 1000 0100 22 1010 31 00 88 2000 0200, 2020 31 00 44 AAA A A E . 1 3, AEB A E 1 3 B AEA 1 1 1000 1000 0100 0100 , 2010 2010 4 33 010 00 3 44 AE 1000 2000 6000 0100 0200 0600 3 2010 2020 6060 4 31 0301 010 00 3 44 B 1 6 2 5 . n n x n y n n x y . 1 1 1 n n x y n n x y 11 11
14、; nn nn xx A yy 2 12 41 , 11 A 3 1 1 1 2 1 2 x y 1 1 n n x y . 1 1 1 52 1 65 6 31 56 nnnn nn n x xx y yx y 1 1 92 10 5 13 10 5 nnn nnn x xy yxy 1 1 92 10 5 13 10 5 nn nn x x yy 92 10 5 . 13 10 5 A 2 12 41 ,5 0 11 12 , . 11 4 , 1 A 1 A 1 1 . 22 1 1 2 , 1 2 2 A A 2 1 2 . 3 11 1 2 11 1 1 2 1 2 nnnnn nnn xxx x AA AA yyy y n A . 12 41 , 11 P 1 1 2 , PAP 1 1 2 , AP P 1 1 1 2 1 41 41 1 11 11 2 11 44 22 15 11 11 4 22 n n n nn nn AP P 1 1 1 1 83 2 1 2 1 10 1 23 2 2 n n n n n x A y 01 pp ,
