第十二章第1节1. (1)%Oz= 5x4-24x3y2,=6y5-12x4y;axay2x3azO22x2y2xln(x? +y2)+一(2)?+y2'axayx?+y2az1azx(3)=X-=+-axayyOzOz= = x[cos(xy)- sin(2xy)];(4)= y[cos(xy) - sin(2xy)] ,axayOz= = e*(xcos y-sin y);(5)=e*(cos y+ xsin y+ sin y),axayx2(μ)z2xOz2(6)2secsecax(ayyJaz1Ozcos=cos+ysin=sin,x1cos=cossin=sin(7)axx2ayyxyx4yyxxyazzxy= y2(1+ xy)-1,(1+xy)(8)=In(1 + xy) +axdy1+xyaz1az1(9)axayx+Inyy(x+lny)O1Oz1(10)-ax1+x21+y2ayouOu=(3x2 + y2 + 22) e(+y2+2) ,+= 2xy e(+*+y+2),(11)axayou = 2xz e(x*+y*+) ,2ylnxInxouQu_ou(12):x=:.OzaxayZ41
第十二章 第 1 节 1. (1) 4 3 2 5x 24x y x z = − ∂ ∂ ， y x y y z 5 4 = 6 −12 ∂ ∂ ； (2) 2 2 3 2 2 2 2 ln( ) x y x x x y x z + = + + ∂ ∂ ， 2 2 2 2 x y x y y z + = ∂ ∂ ； (3) y y x z 1 = + ∂ ∂ ， 2 y x x y z = − ∂ ∂ ； (4) y[ ] cos(xy) sin(2xy) x z = − ∂ ∂ ， x[ ] cos(xy) sin(2xy) y z = − ∂ ∂ ； (5) e (cos y x sin y sin y) x z x = + + ∂ ∂ ， e (x cos y sin y) y z x = − ∂ ∂ ； (6) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∂ ∂ y x y x x z 2 2 sec 2 ， ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ∂ ∂ y x y x y z 2 2 2 2 sec ； (7) x y y x x y z cos cos 1 = ∂ ∂ x y y x x y sin sin 2 + ， x y y x y x y z cos cos 2 = − ∂ ∂ x y y x x sin sin 1 − ； (8) 2 1 (1 ) − = + ∂ ∂ y y xy x z ， ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = + + + ∂ ∂ xy xy xy xy y z y 1 (1 ) ln(1 ) ； (9) x x y z ln 1 + = ∂ ∂ ， ( ln ) 1 y y x y z + = ∂ ∂ ； (10) 2 1 1 x x z + = ∂ ∂ ， 2 1 1 y y z + = ∂ ∂ ； (11) (3 ) 2 2 2 x y z x u = + + ∂ ∂ ( ) 2 2 2 x x y z e + + ， = ∂ ∂ y u 2xy ( ) 2 2 2 x x y z e + + ， = ∂ ∂ z u 2xz ( ) 2 2 2 x x y z e + + ； (12) −1 = ∂ ∂ z y x z y x u ， = ∂ ∂ y u z y x z ln x ， = ∂ ∂ z u z y x z y x 2 ln − ； 1
ouauXOu1(13)axayOz(r2 + y2 + 22)x2-2+ 112ouououii-!-lxyInx,=yxyInxlny;(14)==axayOzOu(15)ai,i=1,2,,n;ax,Ou=zayou=Za(16)i=12,...,n,j=1,2,.,nax;ay,j-=li=l22. f,(3,4)=f,(3,4) =554. 0=45. (1) df(1,2) = 8dx - dy;48dx+(2) df(2,4) =2212)=d(3) df(0,1)= dx, dyS846. (1)dz = y* In ydx + xyx-l dy;(2) dz = e (1 + xy)(ydx + xdy):2y2x(3)dz：dx +dy(x- y)2(x-y)2x2xy(4)dz =dxdy(x2 +y2)(x? +y2)2xdx+ ydy+ zdz(5)du= Vx2 + y2 +222(xdx + ydy + zdz)(6)du=x? + y2 +221Oz>avV22
(13) ( )2 3 2 2 2 x y z x x u + + = − ∂ ∂ ， = ∂ ∂ y u ( )2 3 2 2 2 x y z y + + − ， = ∂ ∂ z u ( )2 3 2 2 2 x y z z + + − ； (14) −1 = ∂ ∂ z z y y x x u ， = ∂ ∂ y u zy x x z z y ln −1 ， = ∂ ∂ z u y x x y z z y ln ln ； (15) a i n x u i i = , = 1,2,", ∂ ∂ ； (16) a y i n x u n j ij j i , 1,2, , 1 = = " ∂ ∂ ∑ = ， a x j n y u n i ij i j , 1,2, , 1 = = " ∂ ∂ ∑ = . 2． 5 2 f x (3,4) = ， 5 1 f y (3,4) = . 4. 4 π θ = . 5. (1) df (1,2) = 8dx − dy ; (2) df dx dy 21 8 21 4 (2,4) = + ; (3) df (0,1) = dx , df dx dy 8 2 8 2 ,2) 4 ( = − π . 6. (1) dz y ydx xy dy ; x x 1 ln − = + (2) dz e (1 xy)( ydx xdy) ; xy = + + (3) dy x y x dx x y y dz 2 2 ( ) 2 ( ) 2 − + − = − ; (4) dx x y xy dz 2 3 2 2 ( + ) = − dy x y x 2 3 2 2 2 ( + ) + ; (5) 2 2 2 x y z xdx ydy zdz du + + + + = ; (6) 2 2 2 2( ) x y z xdx ydy zdz du + + + + = . 7. 2 1 = − ∂ ∂ v z . 2
Oz8.a)=cosα+sina,5元5元T元(1)v = (cos *); (2) V=(cos,sinsin-44443元7元m3元7元或v=(cos(3) v=(cos,sin,sin444A14af(2)9. (1) grad f(1,2) =(2,2),v/a,2)510. (1)grad z =(2x + y cos(xy), 2ysin(xy)+xy2 cos(xy));(2) gad=(--);a3,62(3) grad u(1,1,1) = (11,9,5),11.在(x,y)(0,0)点，增长最快的方向为gradf=(y,x)；在(0,0)点，增长最快的方向为(1,1)和(-1,-1)0?2 -y2 -x216. (1) 0=az2xy2xy(x? + y2)2ax?(x?+y2)2x0y(x2 +y2)2y2a?z(2)(2 - y)cos(x+ y)-xsin(x+ y),=ax?a?z=(1- y)cos(x+ y)-(1+ x)sin(x+ y),axdy02ycos(x+ y)-(x +2)sin(x+ y)ay2a32a3z(2+4xy+x2)e(3)=(3x2+x3y)eyaxoy2axay6a4atua'u6ab2(4)ax4ax?ay?(ax + by + cz)4(ax + by + cz)4 ?ap+9z(5)= plq!.axPay!ap+g+ru=(x + p)(y+q)(z + r)ex+y+:(6)axPayaz"3
8. (1,1) = cosα + sinα ∂ ∂ v z , (1) ) 4 ,sin π π 4 v =(cos ; (2) ) 4 5 ,sin 5π π 4 v =(cos ; (3) ) 4 3 ,sin 3π π 4 v =(cos 或 ) 4 7 ,sin 7π π 4 v =(cos . 9. (1) grad f (1,2) = (2,2), (2) 5 14 (1,2) = ∂ ∂ v f . 10. (1) grad ( 2 cos( ), 2 sin( ) cos( )); 3 2 z = x + y xy y xy + xy xy (2) ) 2 , 2 grad ( 2 2 b y a x z = − − ; (3) grad u(1,1,1) = (11,9,5) . 11. 在 (x, y) ≠ (0,0) 点, 增长最快的方向为 grad f = ( y, x) ; 在 点, 增长最快 的方向为 和 (0,0) (1,1) (−1,−1). 16. (1) 2 2 2 2 2 ( ) 2 x y xy x z + = ∂ ∂ , = ∂ ∂ ∂ x y z 2 2 2 2 2 2 (x y ) y x + − , = ∂ ∂ 2 2 y z 2 2 2 ( ) 2 x y xy + − ; (2) (2 ) cos( ) sin( ) 2 2 y x y x x y x z = − + − + ∂ ∂ , = ∂ ∂ ∂ x y z 2 (1− y) cos(x + y) − (1+ x)sin(x + y), = ∂ ∂ 2 2 y z − y cos(x + y) − (x + 2)sin(x + y). (3) xy xy x y e x y z (2 4 ) 2 2 2 3 = + + ∂ ∂ ∂ , xy x x y e x y z (3 ) 2 3 2 3 = + ∂ ∂ ∂ . (4) 4 4 4 4 ( ) 6 ax by cz a x u + + = − ∂ ∂ , 4 2 2 2 2 4 ( ) 6 ax by cz a b x y u + + = − ∂ ∂ ∂ . (5) p!q! x y z p q p q = ∂ ∂ ∂ + . (6) x y z p q r p q r x p y q z r e x y z u + + + + = + + + ∂ ∂ ∂ ∂ ( )( )( ) . 3
22dxdy-号 dy2;17. (1)d2z = -dx2 + 3yxy(2) d3= = -4 sin 2(ax + by)(adx + bdy)3:(3) d3u = e*+y+*[(x2 + y2 +22 +6x+6)dx3 +(x2 + y2 + 22 +6y+6)dy3+(x? + y? +2? + 6z +6)d-']+3e++y+ [(x? + y? + 2? + 4x+2y+2)dx d)+(x2+y2+=2+4y+2z+2)dydz+(x2+y2+2+4z+2x+2)dz*dx+(x2+y?+22+2x+4y+2)dxdy2+(x2+y2+22+2y+4z+2)dydz2+(x? + y2 +22 +22+ 4x+2)dzdx2] +e*+y+(x2 + y2 + 22 +2x+2y+22)dxdydz .2(si( +号元元dxdyk-i(4) dkz =i218. f(x, y) =(2 - x)sin y- =In(1- xy) + y3..320.α=2ab,o);C21. (1) f ()-2, 21(3 e?-2e2(2) f(1, 2, ") A1216(3 (-1 0)(3) g'(1, 元) =|0 (0 1[fi(x,y,z)=x+C22. (2) f2(x, y,2) = y +C2 ;s(x,y,2)=z+C3[Ji(x, y,2)=[ p(x)dx(3) f2(x, ,z)=Jq(y)dy[Js(x, y,2)=[r(2)dz第2节
17. (1) 2 2 2 1 2 2 dy y x dxdy y dx x d z = + − ; (2) ; 3 3 d z = −4sin 2(ax + by)(adx + bdy) (3) 3 2 2 2 3 d u e [(x y z 6x 6)dx x y z = + + + + + + 2 2 2 3 + (x + y + z + 6y + 6)dy ( 6 6) ] 2 2 2 3 + x + y + z + z + dz + e x y z x y dx dy x y z 2 2 2 2 3 [( + + + 4 + 2 + 2) + + x y z y z dy dz 2 2 2 2 + ( + + + 4 + 2 + 2) x y z z x dz dx 2 2 2 2 + ( + + + 4 + 2 + 2) 2 2 2 2 + (x + y + z + 2x + 4y + 2)dxdy 2 2 2 2 + (x + y + z + 2y + 4z + 2)dydz ( 2 4 2) ] 2 2 2 2 + x + y + z + z + x + dzdx e x y z x y z dxdydz x y z ( 2 2 2 ) 2 2 2 + + + + + + + + . (4) x i k i k i k dx dy k i e y i k d z − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ π 2 sin 0 . 18. 3 ln(1 ) 1 ( , ) (2 )sin xy y y f x y = − x y − − + . 20. 2 3 α = − . 21. (1) ) 4 ( π f' T , ) 2 2 , 2 2 = (− a b c ; (2) ) 4 (1, 2, π f' ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 3 12 16 3 2 2 2 e e ; (3) g'(1, π) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − = 0 1 0 1 1 0 . 22. (2) ; ⎪ ⎩ ⎪ ⎨ ⎧ = + = + = + 3 3 2 2 1 1 ( , , ) ( , , ) ( , , ) f x y z z C f x y z y C f x y z x C (3) . ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = = ∫ ∫ ∫ f x y z r z dz f x y z q y dy f x y z p x dx ( , , ) ( ) ( , , ) ( ) ( , , ) ( ) 3 2 1 第 2 节 4
dz(1)(21.0)sec(2tdtd2[(cost-6t?)-sint-12t];(2)dt?dw(3)=eax sinx;dx3x2Oz2xln(3x-2y)+axy2y2(3x-2y)(4)2x22x2OzIn(3x -2y)-ayy2(3x-2y)LOu(2x+2ysinxcosx)ax(5)Ozer+y+y'sin**(2y+4y sin?x)Oy[=e'(sinu+2xcosu)+e'(sinu+2cosu)+e(sinu+2cos)at(6)aw=te(sinu+2xvcosu)+es+t(sinu+2zvcosu)as其中u=x?+y?+2?,V=x+y+z;%=2(u +V)-sin(u+V+ aresin v),(7)au"= =2-cos(u +v+ arcsin v)1+Ovou%=[.](8)axouXxy,-12ayVaux1J2xy,3axoyyau2x[x,]+xfil)ya2V
1. (1) ) 2 )sec(2 4 (2 3 2 t t dt t dz = − + ; (2) [(cos 6 ) sin 12 ] sin 2 2 2 2 3 e t t t t dt d z t t = − − − − ; (3) e x dx dw ax = sin ; (4) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − = − − − ∂ ∂ − = − + ∂ ∂ (3 2 ) 2 ln(3 2 ) 2 (3 2 ) 3 ln(3 2 ) 2 2 2 3 2 2 2 2 y x y x x y y x y z y x y x x y y x x z ; (5) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = + ∂ ∂ = + ∂ ∂ + + + + (2 4 sin ) (2 2 sin cos ) sin 3 2 sin 4 2 2 4 2 2 2 4 2 e y y x y z e x y x x x u x y y x x y y x ; (6) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = + + + ∂ ∂ = + + + + + ∂ ∂ + + (sin 2 cos ) (sin 2 cos ) (sin 2 cos ) (sin 2 cos ) (sin 2 cos ) te u xv u e u zv u s w e u xv u e u yv u e u zv u t w s s t s t s t 其中u = x 2 + y 2 + z 2，v = x + y + z ; (7) 2(u v) sin(u v arcsin v) u z = + − + + ∂ ∂ , ) 1 1 2 cos( arcsin )(1 2 2 v u v v v u z − = − + + + ∂ ∂ ∂ ; (8) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∂ ∂ y x f xy y y x yf xy x u , 1 , 1 2 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∂ ∂ y x f xy y x y x xf xy y u , , 1 2 2 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∂ ∂ ∂ y x f xy y y x f xy x y u , 1 , 1 2 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + y x f xy y x y x xyf xy, , 11 3 22 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∂ ∂ y x x f xy y x f xy y x y u , , 2 11 2 2 3 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − y x f xy y x y x f xy y x , , 2 4 22 2 2 12 2 . 5