数学分析第三版复旦大学欧阳光中.pdf

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1、214 x2. 1. (1) u =f(x,y)x =rcos,y =rsin u r , 2 u r 2 (2) u =f(x,y)x =a,y =b u , 2 u 2 , 2 u , u , 2 u 2 (3) u =f(x 2 +y 2 +z 2 ) u x , 2 u x 2 , 2 xy , u y , u z (4) u =f x, x y ! u x , 2 u x 2 , u y . (1) u r =fxcos +fy sin 2 u r 2 =f x 2 cos 2 +fxy sin2 +f y 2 sin 2 (2) u =afx, 2 u 2 =a 2 f x 2,

2、2 u =abfxy, u =bfy, 2 u 2 =b 2 f y 2 (3) u x = 2xf 0 (x 2 +y 2 +z 2 ), 2 u x 2 = 2f 0 (x 2 +y 2 +z 2 )+4x 2 f 00 (x 2 +y 2 +z 2 ), 2 xy = 4xyf 00 (x 2 +y 2 +z 2 ), u y = 2yf 0 (x 2 +y 2 +z 2 ), u z = 2zf 0 (x 2 +y 2 +z 2 ) (4) u x =f1 + 1 y f2, 2 u x 2 =f11 + 2 y f12 + 1 y 2 f22, u y = x y 2 f2 2. =

3、 (x,y,z),x =u+v,y =uv,z =uv u,v. u = x +y +vz,v = xy +uz 3. () (1) u =f(x+y) (2) u =f(x+y,xy) (3) u =f(ax 2 +by 2 +cz 2 ) (1) du =f 0 (x+y)(dx+ dy) (2) du = (f1 +f2)dx+(f1f2)dy (3) du = 2f 0 (ax 2 +by 2 +cz 2 )(axdx+bydy +czdz) 4. (1) z =(x 2 +y 2 )y z x x z y = 0 (2) u =y(x 2 y 2 )y u x +x u y = xu

4、 y (3) u =x(x+y)+y (x+y) 2 u x 2 2 2 u xy + 2 u y 2 = 0. (1) z x = 2x 0 (x 2 +y 2 ), z y = 2y 0 (x 2 +y 2 ) y z x x z y = 0. (2) u x = 2xy 0 (x 2 y 2 ), u y =(x 2 y 2 )2y 2 0 (x 2 y 2 ) y u x +x u y =x(x 2 y 2 ) = xu y .215 (3) u x =(x+y)+x 0 (x+y)+y 0 (x+y), u y =x 0 (x+y)+ (x+y)+y 0 (x+y) 2 u x 2

5、= 2 0 (x+y)+x 00 (x+y)+y 00 (x+y), 2 u xy = 0 (x+y)+ 0 (x+y)+x 00 (x+y)+y 00 (x+y) 2 u y 2 = 2 0 (x+y)+x 00 (x+y)+y 00 (x+y) 2 u x 2 2 2 u xy + 2 u y 2 = 0. 5. u = 2 u x 2 + 2 u y 2 + 2 u z 2 ,u =f(x+y +z,x 2 +y 2 +z 2 ). u x =f1 +2xf2 2 u x 2 =f11 +4xf12 +4x 2 f22 +2f2 2 u y 2 =f11 +4yf12 +4y 2 f22

6、 +2f2, 2 u z 2 =f11 +4zf12 +4z 2 f22 +2f2 u = 2 u x 2 + 2 u y 2 + 2 u z 2 = 3f11 +4(x+y +z)f12 +4(x 2 +y 2 +z 2 )f22 +6f2. 6. u =f(r),r = p x 2 +y 2 f (r) 2 u x 2 + 2 u y 2 = d 2 u dr 2 + 1 r du dr u x = x p x 2 +y 2 f 0 (r) 2 u x 2 = y 2 (x 2 +y 2 ) 3 2 f 0 (r)+ x 2 x 2 +y 2 f 00 (r) 2 u y 2 = x 2

7、(x 2 +y 2 ) 3 2 f 0 (r)+ y 2 x 2 +y 2 f 00 (r) 2 u x 2 + 2 u y 2 = d 2 u dr 2 + 1 r du dr 7. u, v x,y x =rcos,y =rsin u x = v y , u y = v x u r = 1 r v , v r = 1 r u . u, v x,y x =rcos,y =rsin u r = cos u x +sin u y , v r = cos v x +sin v y , u =rsin u x +rcos u y , v =rsin v x +rcos v y u x = v y ,

8、 u y = v x u r = 1 r v , v r = 1 r u . 8. f (tx,ty) =t n f(x,y) x f x +y f y =nf n.z = p x 2 +y 2 x z x +y z y . f (tx,ty) =t n f(x,y)tf 1(tx,ty)x+f2(tx,ty)y =nt n1 f(x,y) t = 1f 1(x,y)x+f2(x,y)y =nf(x,y)x f x +y f y =nf z (x,y) = p x 2 +y 2 z (tx,ty) =t p x 2 +y 2 (t 0) x z x +y z y =z = p x 2 +y 2

9、 . 9. z =x y x ! + y x !216 x 2 2 z x 2 +2xy 2 z xy +y 2 2 z y 2 = 0 z x = y x ! y x 0 y x ! y x 2 0 y x ! , z y = 0 y x ! + 1 x 0 y x ! 2 z x 2 = y 2 x 3 00 + 2y x 3 0 + y 2 x 4 00 , 2 z xy = y x 2 00 1 x 2 0 y x 3 00 , 2 z y 2 = 1 x 00 + 1 x 2 00 x 2 2 z x 2 +2xy 2 z xy +y 2 2 z y 2 = 0 10. u =(x+

10、at)+ (xat), 2 u t 2 =a 2 2 u x 2 . u =(x+at)+ (xat), u t =a( 0 0 ), u x = 0 + 0 2 u t 2 =a 2 ( 00 + 00 ), 2 u x 2 = 00 + 00 2 u t 2 =a 2 2 u x 2 .217 x3. () 1. z =f(x,y) (1) x+y +z =e z (2) xyz =x+y +z (1) x1+ zx =zxe z z x = 1 e z 1 z x 2 = e z (1e z ) 3 z y = 1 e z 1 ,z y 2 = e z (1e z ) 3 ,zxy =z

11、yx = e z (1e z ) 3 (2) xy z +xyzx = 1+zx ()z x = yz1 1xy ()x2y zx +xyz x 2 =z x 2 z x 2 = 2yzx 1xy = 2y(yz1) (xy1) 2 z y = xz1 1xy ,z y 2 = 2x(xz1) (xy1) 2 ,zxy =zyx = 2z (xy1) 2 2. (1) f(x+y,y +z,z +x) = 0 z x , z y (2) z =f(xz,zy) dz (3) F(xy,yz,zx) = 0 z x , z y (4) F(x,x+y,x+y +z) = 0 z x , z y

12、, 2 z x 2 . (1) xz =z(x,y)f 1 +f2zx +f3(zx +1) = 0z x = f1 +f3 f2 +f3 z y = f1 +f2 f2 +f3 (2) dz = (xdz +zdx)f1 +(dz dy)f2 dz = zf1dxf2dy 1xf1f2 (3) xz =z(x,y)F 1F2zx +F3(zx1) = 0z x = F1F3 F2F3 z y = F2F1 F2F3 (4) xz =z(x,y)F 1 +F2 +F3(1+zx) = 0 ()z x = F1 +F2 +F3 F3 ()x F11 +F12 +F13(1+zx)+F21 +F2

13、2 +F23(1+zx)+z x 2F3 +(1+zx)F13 +F23 +F33(1+zx) = 0 z x 2 = 1 F 3 3 F 2 3 (F11 +2F12 +F22)2F3(F1 +F2)(F13 +F23)+F33(F1 +F2) 2 z y = F2 +F3 F3 3. z =x+y(z)z =z(x,y)1 y 0 (z)6= 0 z y =(z) z x z =z(x,y) dz = dx+(z)dy +y 0 (z)dz 1 y 0 (z)6= 0 dz = dx+(z)dy 1y 0 (z) z y = (z) 1y 0 (z) , z x = 1 1y 0 (z)

14、z y =(z) z x218 4. ax +by +cz = (x 2 +y 2 +z 2 ) z =z(x,y) (cy bz) z x +(azcx) z y = bxay (u)ua, b,c. z =z(x,y)(u)u a dx+bdy +cdz = 2(xdx+ydy +zdz) 0 z x = 2x 0 a c2z 0 , z y = 2y 0 b c2z 0 (cy bz) z x +(azcx) z y =bxay 5. (cx az,cybz) = 0z =z(x,y)a z x +b z y =c. x, y z =z(x,y) c1a1zxb2zx = 0,a1zy

15、+c2b2zy = 0 z x = c1 a1 +b2 , z y = c2 a1 +b2 a z x +b z y =c. 6. F (x+zy 1 ,y +zx 1 ) = 0z =z(x,y)x z x +y z y =zxy. x, y z =z(x,y) F1 1+ zx y ! +F2 zx x z x 2 ! = 0,F1 zy y z y 2 ! +F2 1+ zy x ! = 0 z x = yzF2x 2 yF1 x(xF1 +yF2) , z y = xzF1xy 2 F2 y(xF1 +yF2) x z x +y z y =zxy. 7. (1) x+y +z = 0,

16、 xyz = 1, dy dx , dz dx , d 2 y dx 2 (2) 8 : x+y =u+v, x y = sinu sinv , du, dv (3) xu+yv = 0, yu+xv = 1, u x , u y , v x , v y , 2 u xy (4) 8 : 1+ dy dx + dz dx = 0 yz +xz dy dx +xy dz dx = 0 () dy dx = y(zx) x(yz) , dz dx = z(xy) x(yz) ()x 8 : d 2 y dx 2 + d 2 z dx 2 = 0 z dy dx +y dz dx +z dy dx

17、+x dy dx dz dx +xz d 2 y dx 2 +y dz dx +x dy dx dz dx +xy d 2 z dx 2 = 0 d 2 y dx 2 = 2z dy dx +2y dz dx +2x dy dx dz dx x(yz) dy dx , dz dx d 2 y dx 2 = yz(xy) 2 +(xz) 2 +(yz) 2 x 2 (zy) 3219 (2) u+v =x+y ysinu =xsinv du+ dv = dx+ dy sinudy +ycosudu = sinvdx+xcosvdv du = 1 xcosv +ycosu (sinv +xcosv

18、)dx(sinuxcosv)dy dv = 1 xcosv +ycosu (sinvycosu)dx+(sinu+ycosu)dy (3) xdu+ydv =udxvdy ydu+xdv =vdxudy du = 1 x 2 y 2 (yvxu)dx+(yuxv)dy, dv = 1 x 2 y 2 (yuxv)dx+(yvxu)dy u x = yvxu x 2 y 2 , u y = yuxv x 2 y 2 , v x = yuxv x 2 y 2 , u x = yvxu x 2 y 2 2 u xy = (yuxvxvx)(x 2 y 2 )2x(yuxv) (x 2 y 2 ) 2

19、 u x , v x 2 u xy = 2(x 2 v +y 2 v2xyu) (x 2 y 2 ) 2 (4) x, y x 8 : 1 =sincos x cossin x 0 =sinsin x +coscos x x = cos sin , x = sin cos z x = z x =cotcos = x z z y = y z (5) x 8 : u x =f1 u x +f2 +f3 v x v x =g1 u x 1 ! +2vyg2 v x u x = f2(12vyg2)g1f3 (f11)(2vyg21)g1f3 , v x = g1(f1 +f21) (f11)(2vy

20、g21)g1f3 . 8. x =u+v,y =u 2 +v 2 ,z =u 3 +v 3 z x,y z x , z y . x 2 y = 2uv z = (u+v)(u 2 uv +v 2 ) = x 2 (3yx 2 ) z x = 3 2 (yx 2 ) , z y = 3 2 x. 9. x =rcos,y =rsin 8 : dx dt =y +kx(x 2 +y 2 ) dy dt =x+ky(x 2 +y 2 ) . x,y tr, tx =rcos,y =rsin t 8 : dx dt = cos dr dt rsin d dt dy dt = sin dr dt +rc

21、os d dt x, y, dx dt , dy dt 8 : cos dr dt rsin d dt =rsin +krcosr 2 sin dr dt +rcos d dt =rcos +krsinr 2 dr dt =kr 3 , d dt =1. 10. x =e u cos,y =e u sin 2 z x 2 + 2 z y 2 = 0. x =e u cos,y =e u sin u = ln(x 2 +y 2 ), = arctan y x220 u x = x x 2 +y 2 , u y = y x 2 +y 2 ; x = y x 2 +y 2 , y = x x 2 +

22、y 2 u x = y , u y = x z x = z u u x + z x , z y = z u u y + z y 2 z x 2 = 2 z u 2 u x ! 2 +2 2 z u u x x + 2 z 2 x ! 2 + z u 2 u x 2 + z 2 x 2 2 z y 2 = 2 z u 2 u y ! 2 +2 2 z u u y y + 2 z 2 y ! 2 + z u 2 u y 2 + z 2 y 2 2 u x 2 = x y ! = y x ! = y u y ! = 2 u y 2 2 x 2 = 2 y 2 2 x 2 + 2 y 2 = 2 u

23、x 2 + 2 u y 2 = 0 u x ! 2 + u y ! 2 = x ! 2 + y ! 2 , u x x = u y y 2 z x 2 + 2 z y 2 =e 2u 2 z u 2 + 2 z 2 ! = 0 2 z u 2 + 2 z 2 = 0. 11. x =rcos,y =rsin f (x,y) = (r,)r, 2 f x 2 + 2 f y 2 . f (x,y) = (r,)r, f x x r + f y y r = r f x cos + f y sin = r f x x + f y y = f x rsin + f y rcos = 2 r 2 = 2

24、 f x 2 cos 2 + 2 f xy sin2 + 2 f y 2 sin 2 2 2 =r 2 2 f x 2 sin 2 2 f xy sin2 + 2 f y 2 cos 2 ! f x rcos f y rsin 2 r 2 + 1 r 2 2 2 = 2 f x 2 + 2 f y 2 1 r r 2 f x 2 + 2 f y 2 = 2 r 2 + 1 r 2 2 2 + 1 r r 12. x =e ,y =e ax 2 2 z x 2 +2bxy 2 z xy +cy 2 2 z y 2 = 0(a,b,c). x =e ,y =e = lnx, = lny d dx

25、= 1 x , d dy = 1 y z x = 1 x z , z y = 1 y z 2 z x 2 = 1 x 2 2 z 2 z ! , 2 z y 2 = 1 y 2 2 z 2 z ! , 2 z xy = 1 xy 2 z a 2 z 2 z ! +2b 2 z +c 2 z 2 z ! = 0. 13. =x, =x 2 +y 2 y z x x z y = 0 . z x,y , x,y z , x,y z x = z +2x z , z y = 2y z y z x x z y =y z 221 y 6 0y z x x z y = 0 z = 0. 14. =x, =yx

26、, =zx u x + u y + u z = 0. ux,y,z , x,y,z u , x,y.z u x = u u u , u y = u , u z = u u x + u y + u z = 0 z = 0 15. = x +1y, = x +2y A 2 u x 2 + 2B 2 u xy +C 2 u y 2 = 0(A,B,C AC B 2 0) 2 u = 0 1,2 C 2 +2B+A = 0. ux,y u, x,y u x = u + u , u y =1 u +2 u 2 u x 2 = 2 u 2 +2 2 u + 2 u 2 , 2 u y 2 = 2 1 2

27、u 2 +212 2 u + 2 2 2 u 2 , 2 u xy =1 2 u 2 +(1 +2) 2 u +2 2 u 2 A 2 u x 2 +2B 2 u xy +C 2 u y 2 = 2 u 2 (A+2B1 +C 2 1 )+2 2 u A+B(1 +2)+C12+ 2 u 2 (A+ 2B2 +C 2 2 ) = 0 A 2 u x 2 +2B 2 u xy +C 2 u y 2 = 0 2 u = 0 8 : A+2B1 +C 2 1 = 0 A+2B2 +C 2 2 = 0 A+B(1 +2)+C126= 0 1,2 C 2 +2B+A = 0 16=2 1,2 C 2 +

28、2B+A = 0 1 +2 = 2B C ,11 = A C A +B(1 +2)+C12 = 2 C (ACB 2 )6= 0 A 2 u x 2 + 2B 2 u xy +C 2 u y 2 = 0 = x +1y, = x +2y 2 u = 0 1,2 C 2 +2B+A = 0. 16. w 2 w x 2 + 2 w y 2 = 0 x =(u,v),y = (u,v) u = v , v = u ! . ! x, y x,y u,v w x,y u,v w u = w x u + w y u , w v = w x v + w y v 2 w u 2 = 2 w x 2 u !

29、2 +2 2 w xy u u + 2 w y 2 u ! 2 + w x 2 u 2 + w y 2 u 2 2 w v 2 = 2 w x 2 v ! 2 +2 2 w xy v v + 2 w y 2 v ! 2 + w x 2 v 2 + w y 2 v 2 u = v , v = u 2 u 2 = 2 uv , 2 v 2 = 2 vu , 2 u 2 = 2 uv , 2 v 2 = 2 vu 2 w u 2 , 2 w v 2 2 w u 2 + 2 w v 2 = 2 w x 2 + 2 w y 2 ! u ! 2 + v ! 2 # 2 w x 2 + 2 w y 2 =

30、0 2 w u 2 + 2 w v 2 = 0 w 2 w x 2 + 2 w y 2 = 0 x =(u,v),y = (u,v). 17. =xat, =x+at 2 u t 2 =a 2 2 u x 2 .222 ut, x, t,xu, t,x u t =a u +a u , u x = u + u 2 u t 2 =a 2 2 u 2 2a 2 2 u +a 2 2 u 2 , 2 u x 2 = 2 u 2 + 2 u + 2 u 2 2 u t 2 =a 2 2 u x 2 4a 2 2 u = 0 a 6 0 2 u = 0. 18. u,v w =w(u,v) (1) u =

31、x 2 +y 2 ,v = 1 x + 1 y ,w = lnz(x+y) y z x x z y = (yx)z (2) u =x+y,v = y x ,w = z x 2 z x 2 2 2 z xy + 2 z y 2 = 0 (3) x =u,y = u 1+uv ,z = u 1+uw x 2 z x +y 2 z y =z 2 (4) u = x y ,v =x,w =xzy y 2 z y 2 +2 z y = 2 x (1) du = 2xdx+2ydy, dv = 1 x 2 dx 1 y 2 dy, dw = 1 z dz dx dy dw = w u du+ w v dv

32、 1 z dz dx dy = w u (2xdx+2ydy)+ w v 1 x 2 dx 1 y 2 dy ! dz = 2xz w u z x 2 w v +z ! dx+ 2yz w u z y 2 w v +z ! dy z x , z y z x y 2 y x 2 ! w v = 0 z x y 2 y x 2 ! 6 0 w v = 0. (2) du = dx+ dy, dv = y x 2 dx+ 1 x dy, dw = z x 2 dx+ 1 x dz dw = w u du+ w v dv z x 2 dx+ 1 x dz = w u (dx+ dy)+ w v y x

33、 2 dx+ 1 x dy ! dz = x w u y x w v + z x ! dx+ x w u + w v ! dy z x =x w u y x w v + z x , z y =x w u + w v223 R = z x z y =w(1+v) w v 2 z x 2 2 2 z xy + 2 z y 2 = x z x z y ! y z x z y ! = R x R y = R u u x u y ! + R v v x v y ! = v w(1+v) w v # y x 2 1 x ! = (1+v) 2 x 2 w v 2 = 0 (1+v) 2 x 6 0 2 w

34、 v 2 = 0. (3) x =u,y = u 1+uv ,z = u 1+uw u =x,v = 1 y 1 x ,w = 1 z 1 x du = dx, dv = 1 x 2 dx 1 y 2 dy, dw = 1 x 2 dx 1 z 2 dz dw = w u du+ w v dv 1 x 2 dx 1 z 2 dz = w u dx+ w v 1 x 2 dx 1 y 2 dy ! dz =z 2 1 x 2 w u 1 x 2 w v ! dx+ z 2 y 2 w v dy z x , z y x 2 z 2 w u = 0 xz 6 0 w u = 0. (4) u = x

35、 y ,v =x,w =xzy w y =x z y 1, w y = w u u y + w v v y , u y = x y 2 , v y = 0 w y = x y 2 w u z y = 1 x 1 y 2 w u y 2 z y 2 +2 z y = 2 x + x y 3 2 w u 2 = 2 x x y 3 6 0 2 w u 2 = 0.224 x4. 1. (1) x =asin 2 t,y =bsintcost,z =ccos 2 tt = 4 (2) x 2 +y 2 +z 2 = 6,x+y +z = 0(1, 2,1). (1) x0 = a 2 ,y0 = b

36、 2 ,z0 = c 2 ,x 0 (t0) =a,y 0 (t0) = 0,z 0 (t0) =c t = 4 8 : x a 2 a = z c 2 c y = b 2 8 : x a + z c = 1 y = b 2 a x a 2 ! c z c 2 ! = 0ax cz = 1 2 (a 2 c 2 ). (2) D(F1,F2) D(y,z) (1,2,1) = 2y 2z 1 1 (1,2,1) =6, D(F1,F2) D(z,x) (1,2,1) = 2z 2x 1 1 (1,2,1) = 0, D(F1,F2) D(x,y) (1,2,1) = 2x 2y 1 1 (1,

37、2,1) = 6 (1, 2,1) x+z2 = 0 y =2 x z = 0. 2. x =t,y =t 2 ,z =t 3 x +2y +z = 4. (t 0,t 2 0 ,t 3 0 )x 0 (t0) = 1,y 0 (t0) = 2t0,z 0 (t0) = 3t 2 0 v =f1,2t0,3t 2 0 g n =f1,2,1gv n = 1+4t0 +3t 2 0 = 0t 0 =1,t0 = 1 3 (1, 1,1), 1 3 , 1 9 , 1 27 ! . 3. x =ae t cost,y =ae t sint,z =ae t x 2 +y 2 =z 2 . x,y,z

38、 x 2 +y 2 =z 2 a 2 e 2t cos 2 t+a 2 e 2t sin 2 t =a 2 e 2t =z 2 x 2 +y 2 =z 2 P (x0,y0,z0) v 1 =fx0,y0,z0g P v 2 =fae t 0 (cost0sint0),ae t 0 (sint0 +cost0),ae t 0 g =fx0y0,x0 +y0,z0g x 2 0 +y 2 0 =z 2 0 cos( v1,v2) = v1v2 jv1jjv2j = 2 p 6 = p 6 3 (x, y,z) . 4. (1) x =t 2 ,y =t 3 ,z =t 4 t = 1 (2) x

39、yz = 1,y 2 =x(1, 1,1). (1) x 0 (t0) = 2,y 0 (t0) = 3,z 0 (t0) = 4f2, 3,4g cos = 2 29 p 29,cos = 3 29 p 29,cos = 4 29 p 29. (2) D(F1,F2) D(y,z) (1,1,1) = xz xy 2y 0 (1,1,1) = 2, D(F1,F2) D(z,x) (1,1,1) = xy yz 0 1 (1,1,1) = 1, D(F1,F2) D(x,y) (1,1,1) = yz xz 1 2y (1,1,1) =3f2, 1,3g cos = p 14 7 ,cos

40、= p 14 14 ,cos = 3 14 p 14.225 x5. 1. (1) x =asincos,y =asinsin,z =acos( 0,0) (2) e x z +e y z = 4(ln2, ln2,1) (3) z = 2x 2 +4y 2 (2, 1,12) (4) ax 2 +by 2 +cz 2 +d = 0(x 0,y0,z0). (1) D(y,z) D(,) ( 0 , 0 ) = asincos acossin 0 asin ( 0 , 0 ) =asin 2 0cos0, D(z,x) D(,) ( 0 , 0 ) =a 2 sin 2 0sin0, D(x,

41、y) D(,) ( 0 , 0 ) =a 2 sin0cos0 sin 0cos0 x+sin0sin0y +cos0z =a xasin0cos0 sin0cos0 = yasin0sin0 sin0sin0 = zacos0 cos0 . (2) (l n2,ln2,1)f x = 2,fy = 2,fz =ln16 x +y2ln2z = 0 xln2 1 = yln2 1 = z1 2ln2 . (3) z x(2,1) = 8,zy(2,1) = 8 8x +8yz = 12 x2 8 = y1 8 = z12 1 . (4) (x 0,y0,z0)f x = 2ax0,fy = 2

42、by0,fz = 2cz0 ax 0 x+by0y +cz0z +d = 0 xx0 ax0 = yy0 by0 = zz0 cz0 . 2. z =xy x +3y +z +9 = 0. M 0(x0,y0,z0)n 1 =fy0,x0,1gn 2 =f1,3,1g n 1kn2 y 1 = x 3 = 1 1 (3, 1,3) x+3 1 = y +1 3 = z3 1 . 3. p x+ p y + p z = p a,(a 0)a . P 0(x0,y0,z0) 1 2 p x0 (xx0)+ 1 2 p y0 (yy0)+ 1 2 p z0 (zz0) = 0 p y0z0(xx0)

43、+ p x0z0(yy0)+ p x0y0(zz0) = 0 p ax0, p ay0, p az0 p a( p x0 + p y0 + p z0) =a. 4. x 2 +y 2 =a 2 ,bz =xy . M 0(x0,y0,z0) M 0 n 1 =f2x0,2y0,0g,n2 =fy0,x0,bg cos = n1n2 jn1jjn2j = 2bz0 jaj p a 2 +b 2 .226 x6. 1. u =x 2 xy +y 2 (1, 1)l = (cos,sin). (1) (2) (3) 0 (4) u. u x = 2xy,uy =x+2y u x(1,1) = 1,u

44、y(1,1) = 1 u l =ux(1,1)cos+uy(1,1)sin u l = cos+sin = p 2sin + 4 ! (1) = 4 l = p 2 2 , p 2 2 ! p 2 (2) = 3 4 l = p 2 2 , p 2 2 ! p 2 (3) = 4 , 3 4 l = p 2 2 , p 2 2 ! l = p 2 2 , p 2 2 ! 0 (4) gradu =ux(1,1)i+uy(1,1)j =i+j. 2. u =xyz M (1,1,1)l = (2,1,3). u x =yz,uy =xz,uz =xy (1, 1,1)u x =uy =uz =

45、1 lcos = 2 p 14 ,sin = 1 p 14 ,cos = 3 p 14 u l =ux(1,1,1)cos+uy(1,1,1)cos +uz(1,1,1)cos = 2 7 p 14 gradu =i+j+k. 3. u =x 2 +2y 2 +3z 2 +xy +3x2y6z O (0,0,0)A(1, 1,1). u x = 2x+y +3,uy = 4y +x2,uz = 6z6 O (0,0,0)u x = 3,uy =2,uz =6gradu = 3i2j+6k,jgraduj = 7 A(1, 1,1)u x = 6,uy = 3,uz = 0gradu = 6i+

46、3j,jgraduj = 3 p 5. 4. (1) grad(u+v) =gradu+gradv , (2) grad(uv) =ugradv +vgradu (3) gradF(u) =F 0 (u)gradu .u =u(x,y),v =v(x,y) (1) (u+v) x = u x + v x , (u+v) y = u y + v y grad(u +v) = (u+v) x , (u+v) y ! = u x , v y ! + u x , v y ! =gradu+gradv. (2) (uv) x =v u x +u v x , (uv) y =v u y +u v y gr

47、ad(uv ) = (uv) x , (uv) y ! =v u x , u y ! +u v x , v y ! =ugradv +vgradu. (3) gradF(u) = F x , F y ! = F 0 (u) u x ,F 0 (u) u y ! =F 0 (u) u x , u y ! =F 0 (u)gradu . 5. grad 1 r = r r 3 r = p x 2 +y 2 +z 2 ,r =xi+yj+zk. r x = x r , r y = y r , r z = z r grad 1 r = d dr 1 r ! gradr = 1 r 2 r x i+ r

48、 y j+ r z k ! = 1 r 2 1 r (xi+yj+zk) = r r 3 .232 x2. 1. (1) f =x+y, x 2 +y 2 = 1 (2) f =x2y +2z, x 2 +y 2 +z 2 = 1 (3) f =xyz, 1 x + 1 y + 1 z = 1 a (x 0,y 0,z 0,a 0) (4) f = 1 x + 1 y , x +y = 2 (5) f =xyz, x 2 +y 2 +z 2 = 1,x+y +z = 0. (1) L =x+y +(x 2 +y 2 1) 8 : x1 = p 2 2 y1 = p 2 2 1 = p 2 2

49、8 : x2 = p 2 2 y2 = p 2 2 2 = p 2 2 L x 2 = 2,Lxy = 0,L y 2 = 2 d 2 L p 2 2 , p 2 2 ! = p 2(dx 2 + dy 2 ) : Lx = 1+2x = 0 Ly =2+2y = 0 Lz = 2+2z = 0 L =x 2 +y 2 +z 2 1 = 0 8 : x1 = 1 3 y1 = 2 3 Z1 = 2 3 1 = 3 2 8 : x2 = 1 3 y2 = 2 3 z2 = 2 3 2 = 3 2 L x 2 =L y 2 =L z 2 = 2,Lxy =Lxz =Lyz = 0 d 2 L(x2

50、,y2,z2) = 3(dx 2 + dy 2 + dz 2 ) 0 1 3 , 2 3 , 2 3 ! 3 1 3 , 2 3 , 2 3 ! 3. (3) L =xyz + 1 x + 1 y + 1 z 1 a ! 8 : Lx =yz x 2 = 0 Ly =xz y 2 = 0 Lz =xy z 2 = 0 L = 1 x + 1 y + 1 z 1 a = 0 x =y =z = 3a, = 81a 4 L x 2(3a,3a,3a) =L y 2(3a,3a,3a) =L z 2(3a,3a,3a) = 6a, Lxy(3a,3a,3a) =Lxz(3a,3a,3a) =Lyz(

51、3a,3a,3a) = 3a d 2 L(3a,3a,3a) = 3a(dx+ dy + dz) 2 + dx 2 + dy 2 + dz 2 0 (3a, 3a,3a) 27a 3 . (4) L = 1 x + 1 y +(x+y2)233 8 : Lx = 1 x 2 + = 0 Ly = 1 y 2 + = 0 L =x+y2 = 0 x =y = = 1 L x 2(1,1) =L y 2(1,1) = 2,Lxy(1,1) = 0 d 2 L(1,1) = 2(dx 2 + dy 2 ) 0(1, 1)2. (5) L =xyz +u(x 2 +y 2 +z 2 1)+v(x+y

52、+z) 8 : Lx =yz +2ux+v = 0 Ly =xz +2uy +v = 0 Lz =xy +2uz +v = 0 Lu =x 2 +y 2 +z 2 1 = 0 Lv =x+y +z = 0 8 : x1 = p 6 6 y1 = p 6 6 z1 = p 6 3 u1 = p 6 12 v1 = 1 6 8 : x2 = p 6 6 y2 = p 6 6 z2 = p 6 3 u2 = p 6 12 v2 = 1 6 8 : x3 = p 6 3 y3 = p 6 6 z3 = p 6 6 u3 = p 6 12 v3 = 1 6 8 : x4 = p 6 3 y4 = p 6

53、 6 z4 = p 6 6 u4 = p 6 12 v4 = 1 6 8 : x5 = p 6 6 y5 = p 6 3 z5 = p 6 6 u5 = p 6 12 v5 = 1 6 8 : x6 = p 6 6 y6 = p 6 3 z6 = p 6 6 u6 = p 6 12 v6 = 1 6 d 2 L = 2u(dx 2 + dy 2 + dz 2 )+2(zdxdy +ydxdz +xdydz) (x 1,y1,z1) d 2 L = p 6 6 (dx 2 + dy 2 + dz 2 4dxdy +2dxdz +2dydz) x 2 +y 2 +z 2 = 12x dx+2ydy

54、 +2zdz = 0(x 1,y1,z1) dx+ dy = 2dz x +y+z = 0 dx+ dy+ dz = 0 dx =dy, dz = 0 d 2 L(x1,y1,z1) = p 6 dx 2 0 p 6 6 , p 6 6 , p 6 3 ! p 6 18 (x 3,y3,z3),(x5,y5,z5) p 6 18 (x 2,y2,z2),(x4,y4,z4),(x6,y6,z6) p 6 18 . 2. f =x m y n z p x +y +z =a,a 0,m 0,n 0,p 0,x 0,y 0,z 0. x 0,y 0,z 0f =x m y n z p ln f =m

55、lnx+nlny +plnz ln f f L =mlnx+nlny +plnz +(x+y +za) 8 : Lx = m x + = 0 Ly = n y + = 0 Lz = p z + = 0 L =x+y +za = 0 8 : x = ma m+n+p y = na m+n+p z = pa m+n+p = m+n+p a ma m+n+p , na m+n+p , pa m+n+p ! L x 2 = m x 2 ,Lxy =Lyz =Lxz = 0,L y 2 = n y 2 ,L z 2 = p z 2 , d 2 L = m x 2 dx 2 n y 2 dy 2 p z

56、2 dz 2 ! 0 ma m+n+p , na m+n+p , pa m+n+p ! ln f f m m n n p p (m+n+p) m+n+p a m+n+p f = x m n n z p (x, y,z) x+y =a z = 0 x+z =a y = 0 y +z =a x = 0 f ! 0 f . 3. x 2 +3y 2 = 12. x 2 (2 p 3) 2 + y 2 4 = 1234 (0, 2) (x, y) (x,y 0)2xy +2 A(0, 2),B(x,y),C(x,y) A(0, 2),B(x,y),C(x,y) S =x(y +2) (x,y )x 2

57、 +3y 2 = 12 S =x(y +2) x 2 +3y 2 = 12(x,y 0) L =x(y +2)+(x 2 +3y 2 12) 8 : x = 3 y = 1 = 1 2 A(0, 2),B(3,1),C(3,1)A(0, 2),B(3,1),C(3,1) (0, 2),(3,1),(3,1)(0, 2),(3,1),(3,1) 9. 4. y 2 = 4xx y +4 = 0. (x,y )d = 1 p 2 jxy +4jy 2 = 4x x y +4 = 0 x y +4 0 y 2 = 4x(x,y )d = 1 p 2 (xy +4) L = 1 p 2 (xy +4)

58、+(y 2 4x) 8 : Lx = 1 p 2 4 = 0 Ly = 1 p 2 +2y = 0 L =y 2 4x = 0 8 : x = 1 y = 2 = 1 4 p 2 L x 2 = 0,L y 2 = 1 2 p 2 ,Lxy = 0, d 2 L(1,2) =L x 2 dx 2 +2Lxy dxdy +L y 2 dy 2 = dy 2 2 p 2 0 (1, 2)(1, 2). 5. z =x 2 +y 2 x +y +z = 1. d = p x 2 +y 2 +z 2 z =x 2 +y 2 ,x+y +z = 1 u =x 2 +y 2 +z 2 L =x 2 +y

59、2 +z 2 +(zx 2 y 2 )+(x+y +z1) 8 : Lx = 2x2x+ = 0 Ly = 2y2y + = 0 Lz = 2z + = 0 L =zx 2 y 2 = 0 L =x+y +z1 = 0 8 : x1 = 1+ p 3 2 y1 = 1+ p 3 2 z1 = 2 p 3 1 = 5 p 3+9 3 1 =7+ 11 3 p 3 , 8 : x2 = 1 p 3 2 y2 = 1 p 3 2 z2 = 2+ p 3 2 = 5 p 3+9 3 2 =7 11 3 p 3 d(x 1,y1,z1) = p 95 p 3,d(x2,y2,z2) = p 9+5 p

60、3 1+ p 3 2 , 1+ p 3 2 ,2+ p 3 ! p 9+5 p 3 1+ p 3 2 , 1+ p 3 2 ,2 p 3 ! p 95 p 3. 6. (a, b,c)Ax +By +Cz +D = 0. (x,y,z )Ax +By+Cz+D = 0(a,b,c)d = p (xa) 2 +(yb) 2 +(zc) 2 (x,y,z )Ax +By +Cz +D = 0 d 0d()d 2 d 2 = (xa) 2 +(yb) 2 +(zc) 2 Ax +By +Cz +D = 0 L = (xa) 2 +(yb) 2 +(zc) 2 +(Ax+By +Cz +D)235 8

61、 : Lx = 2(xa)+A = 0 Ly = 2(yb)+B = 0 Lz = 2(zc)+C = 0 L =Ax+By +CzD = 0 8 : x =a 1 2 A y =b 1 2 B z =c 1 2 C = 2(Aa+Bb+Cc+D) A 2 +B 2 +C 2 d = jAa+Bb+Cc+Dj p A 2 +B 2 +C 2 x, y,z 1d !1 (x,y,z) (xa) 2 +(yb) 2 +(zc) 2 d (a,b,c)Ax +By +Cz +D = 0d = jAa+Bb+Cc+Dj p A 2 +B 2 +C 2 .264 x1. 1. Z l (x+y)dsl

62、O (0,0),A(1,0),B(0,1). I = Z l (x+y)ds = Z OA + Z AB + Z BO (x+y)ds OA y = 0, ds = dx Z OA (x+y)ds = Z 1 0 xdx = 1 2 AB y = 1x, ds = p 2 dx Z AB (x+y)ds = Z 1 0 p 2dx = p 2 BO x = 0, ds = dy Z BO (x+y)ds = Z 1 0 ydy = 1 2 I = 1+ p 2 . 2. Z l (x 2 +y 2 )dsl R . l :x =Rcos,y =Rsin, 2 66 3 2 ds = p x 2

63、 +y 2 d =Rd Z l (x 2 +y 2 )ds =R 3 . 3. Z l (x 2 +y 2 +z 2 )dsl x =acost,y =asint,z =bt (06t6 2). ds = p x 02 (t)+y 02 (t)+z 02 (t) dt = p a 2 +b 2 dt I = Z l (x 2 +y 2 +z 2 )ds = 2 3 (3a 2 +4 2 b 2 ) p a 2 +b 2 . 4. Z l x 2 dsl x 2 +y 2 +z 2 =a 2 x +y +z = 0. Z l x 2 ds = Z l y 2 ds = Z l x 2 ds Z

64、l x 2 ds = 1 3 Z l (x 2 +y 2 +z 2 )ds = a 2 3 Z l ds = 2 3 a 3 . 5. Z l z 2 x 2 +y 2 dsl x =acost,y =asint,z =at,(06t6 2). ds = p x 02 (t)+y 02 (t)+z 02 (t) dt = p 2 dtI = Z l z 2 x 2 +y 2 ds = 8 p 2 3 3 a. 6. l x =e t cost,y =e t sint,z =e t ,(06t6t0) (1, 0,1)1. = k x 2 +y 2 +z 2 (1, 0,1) = 1k = 2

65、= 2 x 2 +y 2 +z 2 =e 2t ds = p x 02 (t)+y 02 (t)+z 02 (t) dt = p 3e t dtm = Z l ds = p 3(1e t 0 ). 7. x =acost,y =bsint(0 6t6 2)M (x,y) =jyj. M = Z l jyjdsl x =acost,y =bsint(06t6 2) (1) ab ds = p x 02 (t)+y 02 (t) dt =a p 1 2 1 cos 2 t dt 1 = p a 2 b 2 a M = Z l jyjds = Z 0 absint p 11cos 2 t dt+ Z 2 a(bsint) q 1 2 1 cos 2 t dt = 2ab q 1 2 1 + 2ab 1 arcsin1 = 2b 2 + 2ab 1 arcsin1 (2) a