高等数学第九章多元函数微分法及其应用

111ÊÊÊÙÙÙõõõ¼¼¼êêꇇ‡©©©{{{999ÙÙÙAAA^^^§3444†††ëëëYYY1.¦e4µ£1¤limx→1y→0ln(x+ey)√x2+y2):4=ln(1+1)√1=ln2£2¤limx→0y→02−√xy+4xy¶)µ4¸lim(x,y)→(0,0)−xyxy(2+√xy+4)=−14£3¤limx→0y→0xy√xy+1−1¶)µ4=lim(x,y)→(0,0)xy(√xy+1+1)xy=2£4¤limx→2y→0sin(xy)y¶):4=lim(x,y)→(2,0)sinxyxy∙x=2£5¤limx→0y→01−cos(x2+y2)(x2+y2)ex2y2.)µ4=lim(x,y)→(0,0)12(x2+y2)2(x2+y2)ex2y2=02©y²e4Ø3£1¤limx→0y→0x+yx−y¶)µ÷y=0%C(0,0)ž4¬÷x=0%C(0,0)ž4−1¤±4Ø3©£2¤limx→0y→0x2y2x2y2+(x−y)2.)µ÷y=x%C(0,0)ž4¬÷y=0%C(0,0)ž40¤±4Ø3©3.^½Ây²µlimx→0y→0xy√x2+y2=0.yµé∀ε0xypx2+y2−0≤xy√2|x||y|=1√2|x||y|ε=|x||y|√2εÏd-δ=p√2ε§K|x−0|δ,|y−0|δ,…(x,y)6=(0,0)ž|f(x,y)−0|εy7§2   êêꆆ†‡‡‡©©©1.¦e¼ê êµ£1¤z=xsin(x+y)+cos2(xy)¶)µzx=xcos(x+y)+sin(x+y)−ysin2xyzy=xcos(x+y)−2xcosxysinxy=x(cos(x+y)−sin2xy)£2¤u=arctan(x−y)z.)µux=11+(x−y)2zz(x−y)z−1=z(x−y)z−1(x−y)2z+1;uy=11+(x−y)2zz(x−y)z−1(−1)=−z(x−y)z−1(x−y)2z+1;uz=11+(x−y)2z(x−y)zln(x−y)=(ln(x−y))(x−y)z(x−y)2z+12©W˜£1¤­‚(z=x2+y24y=43:(2,4,5)?ƒ‚†x¶•¤¤–π4.£2¤f(x,y)=x+(y−1)arcsinqxy§Kf0x(x,1)=1.£3¤z=e−(1x+1y)§Kx2∂z∂x+y2∂z∂y=2z.3©f(x,y)=(x2y2(x2+y2)3/2,(x,y)6=(0,0)0,(x,y)=(0,0)^½Ây²µf(x,y)3(0,0)?ëY§… ê3.y²µé∀ε0,‡¦x2y2(x2+y2)3/2−0≤14x2+y2
2(x2+y2)3/2=14px2+y2ε-δ=4ε,Ké∀P(x,y)∈U0(O,δ)k|f(P)−0|ε=lim(x,y)→(0,0)f(P)=f(P0)§Ïdf(x,y)3(0,0)?ëY´f0x(0,0)=f0y(0,0)=0.4©¦e¼ê ê∂2z∂x2,∂2z∂y2Ú∂2z∂x∂yµ£1¤z=arctanxy¶)µzx=yx2+y2,zy=−xx2+y2;zxx=−2xy(x2+y2)2,zxy=x2−y2(x2+y2)2,zyy=2xy(x2+y2)2£2¤z=ylnx.)µzx=1xylnxlny,zy=ylnx−1lnxzxx=1x2ylnx(lny)(lny−1)zxy=1xy−1+lnx(lnxlny+1)zyy=y−2+lnx(lnx)(lnx−1)85©z=xln(xy)§¦∂3z∂x∂y2.)µzx=lnxy+1,zxy=1y,zxyy=−1y26.ä£1¤e¼êz=f(x,y)3:P0Œ‡§K¼ê3:P0 ê3.£X¤£2¤ ê3´Œ‡¿©^‡.(פ£3¤Œ‡7ëY.£X¤£4¤ëY7Œ‡.£×¤£5¤e¼ê3˜: ê3…ëY§K¼ê3T:˜½Œ‡.£X¤7©¦e¼ê‡©µ£1¤z=(1+xy)y;)µdz=y2(1+xy)y−1dx+(1+xy)y[ln(1+xy)+yx1+xy]dy£2¤z=2y√x2+y2¶)µdz=−2xy(x2+y2)32dx+2x2(x2+y2)32dy£3¤u=(xy)z.):dz=yz(xy)z−1dx+xz(xy)z−1dy8©|^‡©/ªØC5¦¼êz=ln(4+x2+y2) ꧿¦dz|x=1,y=2Š.):dz=14+x2+y2d(4+x2+y2)=14+x2+y2(2xdx+2ydy)=2x4+x2+y2dx+2y4+x2+y2dyzx=2x4+x2+y2,zy=2y4+x2+y2dz|x=1,y=2=29dx+49dy9©?ؼêf(x,y)=((x2+y2)sin1x2+y2x2+y26=0,0x2+y2=0,3(0,0):Œ‡5.):Ϗfx(0,0)=fy(0,0)=0limρ→0Δf−0∙Δx−0∙Δyρ=lim(x,y)→(0,0)x2+y2
sin1x2+y2px2+y2=lim(x,y)→(0,0)px2+y2sin1x2+y2=0¤±f(x,y)3(0,0)?Œ‡10©OŽp(1.02)3+(1.97)3CqŠ.)µ-u(x,y)=px3+y3,(x0,y0)=(1,2),Δx=0.02,Δy=−0.03f(P)−f(P0)≈du|(1,2)f(P)=2.959§3õõõEEEÜÜܼ¼¼êêꆆ†ÛÛÛ¼¼¼êêꦦ¦1.¦)eˆKµ£1¤z=ex−2y,,x=sint,y=t3§¦dzdt¶):dzdt=ex−2ycost−2ex−2y∙3t2=esint−2t3cost−6t2esint−2t3£2¤z=xxy§¦∂z∂x,∂z∂y¶)µ∂z∂x=xy+xy−1(ylnx+1),∂z∂y=xy+xyln2x£3¤z=f(x2−y2,exy)§¦∂z∂y¶)µ∂z∂y=f01∙(−2y)+f02∙(xexy)£4¤u=f(xy,yz)§¦∂u∂y¶)µ∂u∂y=f01∙−xy2+f02∙1z£5¤u=f(x,xy,xyz)§¦∂u∂x,∂u∂z.)µ∂u∂x=f01+f02∙y+f03yz∂u∂z=f03∙xy2©¦e¼ê êµ£1¤z=f(x2−y2)§¦∂2z∂y2¶)µzy=f0∙(−2y),zyy=−2f0+4y2f00£2¤z=f(y,xy)§¦∂2z∂y∂x¶)µzy=f01−xy2f02zyx=1yf0012−1y2f02−xy3f0022£3¤z=x3f(xy,yx)§¦∂2z∂x∂y¶)µzx=3x2f+x3f01∙y−f02∙yx2zxy=3x2f01∙x+f02∙1x+x3f01+x3yf0011∙x+f0012∙1x−xf02−xyxf0021+1xf0022=4x3f01+2xf02+x4yf0011−yf0022£4¤z=f(u,x,y),u=xey§¦∂2z∂x∂y.)µzx=f01∙ey+f02zxy=eyf01+ey(f0011∙xey+f0013)+f0021∙xey+f0023103.®f(x,x2)=x4+2x3+x§f01(x,x2)=2x2−2x+1¦f02(x,x2).)µòf(x,x2)=x4+2x3+xüéx¦,f01x,x2
+f02x,x2
∙2x=4x3+6x2+12òf01(x,x2)=2x2−2x+1“\þª§2x2−2x+1+f02x,x2
∙2x=4x3+6x2+1Ïdf02x,x2
=2x2+2x+14©¼êz=f(x,y)÷v§y∂z∂x−x∂z∂y=0§-ξ=x,η=x2+y2(y6=0)¦yµ∂z∂ξ=0.y²µzx=zξ+2xzη,zy=2yzη§Ïd0=y∂z∂x−x∂z∂y=y(zξ+2xzη)−x(2yzη)=zξ5.¦)eˆKµ£1¤x+2y+z−2√xyz=0§¦∂z∂x,∂z∂y)µüéx¦1+z0x−2∙121√xyz(yz+xyz0x)=0u´z0x=−yz−√xyzxy−√xyzüéy¦2+z0y−2∙121√xyzxz+xyz0y
=0z0y=−1xy√xyz−1xz√xyz−2=−xz−2√xyzxy−√xyz£2¤x2a2+y2b2+z2c2=1§¦∂2z∂x∂y.)µ-F=x2a2+y2b2+z2c2−1,u´F0x=2xa2,F0y=2yb2,F0z=2zc2∂z∂x=−F0xF0z=−c2xa2z,∂z∂y=−F0yF0z=−c2yb2z∂2z∂x∂y=−c2xa2∙−1z2∙z0y=−c2xa2−1z2−c2yb2z=−c4a2b2xyz3£3¤x2+y2+z2=xf(yx)§fŒ‡§¦∂z∂x¶):-F=x2+y2+z2−xf(yx)F0x=2x−f+yxf0,F0z=2zu´,∂z∂x=−F0xF0z=−2x−f+yxf02z=xf−yf0−2x22xz6.x2+y+z+f(xyz)=Fx2,y2+z2
.Ù¥f(x),F(x)Œ‡§¦∂z∂x,∂z∂y.)§üéx¦ ê§2x+z0x+f0∙(yz+xyz0x)=2xF01+2z∙z0xF02u´z0x=2xF01−yzf0−2x1−2zF02+xyf0Ón§§üéy¦ ê§z0y=1+xzf0−2yF021−2zF02+xyf0117©ey=y(x),z=z(x)d§|(x+y+z+z2=0x+y2+z+z3=0(½§¦dydx,dzdx.):éx¦(1+y0+z0+2zz0=01+2yy0+z0+3z2z0=0u´y0=−11+2z11+3z211+2z2y1+3z2=−3z2−2z3z2−4yz−2y+1z0=−112y111+2z2y1+3z2=2y−13z2−4yz−2y+18©(x=eu+usinvy=eu−ucosv§¦∂u∂x,∂v∂y.):éx¦ ê((eu+sinv)u0x+(ucosv)v0x=1(eu−cosv)u0x+(usinv)v0x=0u´u0x=1ucosv0usinveu+sinvucosveu−cosvusinv=sinveusinv−(cosv)eu+1aq/,Œv0y=eu+sinvu(eu(sinv−cosv)+−1)9©y=f(x,t),t=t(x,y)÷v§F(x,y,t)=0§f,FÑk˜ëY ê.y²µdydx=f0xF0t−f0tF0xF0t+f0tF0y.y²:§y=f(x,t(x,y))üéx¦y0=f0x+f0t∙t0x+t0yy0
y0=f0x+f0t∙t0x1−f0t∙t0y=f0x+f0t∙−F0xF0t1−f0t∙−F0yF0t=f0xF0t−f0tF0xF0t+f0tF0y10©z=f(u)§¼êu=u(x,y)d§u=ϕ(u)+RxyP(t)dt(½§eϕ,fь‡§PëY¼ê§y²µP(y)∂z∂x+P(x)∂z∂y=0.y²:Äk§u=ϕ(u)+RxyP(t)dtüéx¦ u0x=ϕ0∙u0x+P(x)u0x=P(x)1−ϕ0§u=ϕ(u)+RxyP(t)dtü2éy¦ u0y=ϕ0∙u0y−P(y)u0y=−P(y)1−ϕ0,z0x=f0∙u0x=f0∙P(x)1−ϕ0,z0y=f0∙u0y=−f0∙P(y)1−ϕ0ÏdP(y)∂z∂x+P(x)∂z∂y=P(y)f0∙P(x)1−ϕ0+P(x)∙−f0∙P(y)1−ϕ0=0y12§4õõõ¼¼¼êêꇇ‡©©©ÆÆÆAAAÛÛÛAAA^^^!!!•••ê