数学分析复旦第二版(陈传璋)ch 13-22.pdf

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1、2041oC1?41n?4Y1.:81.y(xn,yn)(x0,y0)xn x0,yn y0(n )y limnMn=M0K 0,N Z+?n Nkr(Mn,M0)=p(xn x0)2+(yn y0)2 uk|xn x0|6 r(Mn,M0),|yn y0|6 r(Mn,M0)|xn x0|2+|yn y0|2=0 6p|xn x0|2+|yn y0|26|xn x0|+|yn y0|qxn x0,yn y0(n )Kp|xn x0|2+|yn y0|2 0(n )=(xn,yn)(x0,y0)(n )2.ye:MnKk4.y limnMn=M0bqk limnMn=M00d 0,N Z+?n

2、 Nkr(Mn,M0)2,r(Mn,M00)2dnkr(M0,M00)6 r(Mn,M0)+r(Mn,M00)0,N Z+?n Nkr(Mn,M0)Kknk nK=nN Ng,kr(Mnk,M0).:(1)Edvy x2:|(2)Edv1 6 x2+y24 4:|(3)Edv0 x2+y2 1:|(4)Edk:(x,y)|xykn.)(1)vy x2:(x,y)E?:vy=x2:(x,y)E.:.(2)v1 x2+y24 4:(x,y)ES:vx2+y24 4:(x,y)E?:vx2+y24=1x2+y24=4:(x,y)E.:.(3)v0 x2+y2 1:(x,y)E?:?:9vx2+y2=

3、1:(x,y)E.:.(4)dkn9n5k:(x,y)E.:.5.yeM0:8E:K3E3:Mn M0(n ).yM0:8E:n=1n3O(M0,1)3E:M16=M03O(M0,2)3E:M2,M26=Mi(i 6=0,1)Xd?1e:Mn(Mn6=Mi)(i=0,1,n 1)r(M0,Mn)0,N Z+?n,m Nkr(Mn,M0)2,r(Mm,M0)2205dlnr(Mm,Mn)6 r(Mn,M0)+r(Mm,M0)0,N Z+?n,m Nkr(Mn,Mm)MnOKIxn,yn|xm xn|r(Mm,Mn),|ym yn|r(Mm,Mn)0y 6 1(2)vy 6 x+1:8(3)x+y

4、 0,0?|x a|,|y b|(x a)2+(y b)26=0k|f(x,y)A|y30|x a|x3-y b=|(x)A|6 ylimxa(x)=Aulimxalimybf(x,y)=limxa(x)=A=limyaxbf(x,y)4.(1)g4f(2)kg43f(3)43?g43f.)(1)f(x,y)=x yx+y,x+y 6=00,x+y=03:(0,0)g4limx0limy0f(x,y)=limx0 xx=1,limy0limx0f(x,y)=limy0 yy=1Klimx0limy0f(x,y)6=limy0limx0f(x,y).(2)f(x,y)=xsin1x+yx+y3:

5、(0,0)g4limy0limx0f(x,y)=limy0yy=1?limx0limy0f(x,y)=limx0 xsin1xx=limx0sin1x3.(3)f(x,y)=xsin1y,y 6=00,y=03:(0,0)g440 6|f(x,y)|=?xsin1y?6|x|K limx0y0f(x,y)=0=43limy0limx0f(x,y)=0?y 0 xsin1y43=limx0limy0f(x,y)3.5.?e3:(0,0)g44(1)f(x,y)=x2y2x2y2+(x y)2(2)f(x,y)=(x+y)sin1x sin1y)(1)limx0limy0f(x,y)=0,limy

6、0limx0f(x,y)=0eUy=kx 04Kk limy=kxx0f(x,y)=limx0 x2k2x2k2+(1 k)2AOOk 6=19k=1B4091d limx0y0f(x,y)3.(2)0 6|f(x,y)|6|x+y|6|x|+|y|K limx0y0f(x,y)=0=43qlimy0(x+y)sin1x sin1y3?x 6=1k!(k=1,2,)limx0(x+y)sin1x sin1y3?y 6=1k!(k=1,2,)=limx0limy0f(x,y)9limy0limx0f(x,y)3.2086.?eY(1)u=1px2+y2(2)u=ln(1 x2 y2)(3)u=1

7、sinxsiny(4)u=ln1(x 1)a+(y b)2+(z c)2)(1)u=1px2+y23:(0,0)?:(0,0)dY:d:?Y(2)S:=vx2+y2 1:u=ln(1 x2 y2)Y:(3)Yx 6=m,y 6=n(m,n=0,1,2,).(4):(a.b.c)?Y.7.yf(x,y)=2xyx2+y2,x2+y26=00,x2+y2=0OuzCxyY?uCY.yky=a 6=0Kxg(x)=f(x,a)=2axx2+a2(x +)?kknq?y=0f(x,0)0w,Yu?Cyf(x,y)uCxYny?Cxf(x,y)uCyY?f(x,y)3:(0,0)?:Y?3:(0,0)Y

8、?:P(x,y)y=mxu?:k limy=mxx0f(x,y)=limx02mx2(1+m2)x2=2m1+m2mK443u limx0y0f(x,y)6=f(0,0)KuC3(0,0):YluCY.8.yf(x,y)=x2yx4+y2,x2+y26=00,x2+y2=03(0,0):zx=tcos,y=tsin(0 6 t+)Y?3(0,0):Y.y?sin=0cos=11u?t 6=0f(tcos,tsin)=0f(0,0)=0Kklimt0f(tcos,tsin)=f(0,0)?sin 6=0klimt0f(tcos,tsin)=0klimt0f(tcos,tsin)=f(0,0)g:

9、P(x,y)y=x2u?:limy=x2x0f(x,y)=126=f(0,0)Kf(x,y)3:(0,0)Y.9.ef(x,y)3,GSCxYCyvoF=?(x,y0)G,(x,y00)Gk|f(x,y0)f(x,y00)|6 L|y0 y00|LKd3GSY.yf(x,y)3GSCxYKGS?:(x0,y0)0?|xx0|1k|f(x,y0)f(x0,y0)|2qf(x,y)3GSyvoFK?(x,y)G,(x,y0)Gk|f(x,y)f(x,y0)|6 L|y y0|209-L|y y0|2K|y y0|2L=min 1,2L!?|x x0|,|y y0|k|f(x,y)f(x0,y0)|

10、6|f(x,y)f(x,y0)|+|f(x,y0)f(x0,y0)|=d3GSY.2101o1.Vg1.e(1)z=x2ln(x2+y2)(2)u=exy(3)z=xy+xy(4)u=arctanyx(5)u=x2+y2+z2+2xy+2yz+2zx(6)u=ecos(+)(1)zx=2xln(x2+y2)+x2x2+y2#,zy=2x2yx2+y2.(2)ux=yexy,uy=xexy.(3)zx=y+1y,zy=x(y2 1)y2.(4)ux=yx2+y2,uy=xx2+y2.(5)ux=2(x+y+z),uy=2(x+y+z),uz=2(x+y+z).(6)u=ecos(+)sin(+)

11、,u=esin(+)+cos(+).2.f(x,y)=x2y2 2yfx(x,y),fy(x,y),fx(2,3),fy(0,0),fy(x,y)?x=yy=x.)fx(x,y)=2xy2,fy(x,y)=2x2y 2,fx(2,3)=36,fy(0,0)=2,fy(x,y)?x=yy=x=2xy2 23.z=ln(x+y)yxzx+yzy=12.yz=ln(x+y)Kzx=12x(x+y),zy=12y(x+y)uxzx+yzy=12.4.e3:(x0,y0)(1)u=x4+y4 4x2y2,(0,0),(1,1)(2)u=xpx2+y2,(1,0),(0,1)(3)u=xsin(x+y),

12、(0,0),4,4!(4)u=ln(x+y2),(0,1),(1,1)(1)du=4x(x2 2y2)dx+4y(y2 2x2)dyK3(0,0):du=03(1,1):du=4dx 4dy.(2)du=y2(x2+y2)32dx xy(x2+y2)32dyK3(1,0):du=03(0,1):du=dx.(3)du=sin(x+y)+xcos(x+y)dx+xcos(x+y)dyK3(0,0):du=03 4,4!:du=dx.211(4)du=dxx+y2+2yx+y2dyK3(0,1):du=dx+2dy3(1,1):du=dx2+dy.5.e(1)u=sin(x2+y2)(2)u=xm

13、 yn(3)u=exy(4)u=xy(5)u=px2+y2+z2(6)u=ln(x2+y2+z2)(1)du=2cos(x2+y2)(xdx+y dy)(2)du=xm1yn1(my dx+nxdy)(3)du=exy(y dx+xdy)(4)du=xy1(y dx+xlnxdy)(5)du=xdx+y dy+z dzpx2+y2+z2(6)du=2(xdx+y dy+z dz)x2+y2+z26.yf(x,y)=p|xy|3(0,0)Yfx(0,0),fy(0,0)3?3(0,0):.yd limx0y0p|xy|=0 limx0y0f(x,y)=0=f(0,0)Kf(x,y)3(0,0):

14、Yfx(0,0)=limx0f(x,0)f(0,0)x=0,fy(0,0)=limy0f(0,y)f(0,0)y=0Kfx(0,0),fy(0,0)3?f(x,y)=p|xy|3(0,0):.eKkf=fx(0,0)x+fy(0,0)y+o()=f=o():P(x,y)y=xu0kf=p|xy|px2+y2=126 0(0)gubKf(x,y)3(0,0):.7.yf(x,y)=xypx2+y2,x2+y26=00,x2+y2=03(0,0):Yfx(x,y),fy(x,y)k.?3(0,0):.yduxypx2+y2?3S7YKf(x,y)3x2+y26=0Yq0?xypx2+y2?=|xy

15、|px2+y26px2+y22,f(0,0)=0K limx0y0f(x,y)=f(0,0)uf(x,y)3(0,0):Ylf(x,y)3(0,0):?SYfx(0,0)=limx0f(x,0)f(0,0)x=0,fy(0,0)=limy0f(0,y)f(0,0)y=0?x2+y26=0fx(x,y)=y3(x2+y2)32,|fx(x,y)|=?y3(x2+y2)32?6 1Kfx(x,y)k.nfy(x,y)k.?f(x,y)3(0,0):.eKkf=fx(0,0)x+fy(0,0)y+o()=f=o():P(x,y)y=xu0kf=xyx2+y2px2+y2=126 0(0)gubKf(

16、x,y)3(0,0):.2128.f(x,y)=(x2+y2)sin1x2+y2,x2+y26=00,x2+y2=0yfx(x,y),fy(x,y)3?Y3(0,0):?.?3(0,0):.y?x2+y26=0fx(x,y)=2xsin1x2+y22xx2+y2cos1x2+y2fx(0,0)=limx0f(x,0)f(0,0)x=0Kfx(0,0)3?3:12n,0!fx 12n,0!=22n (n )fx(x,y)3(0,0):?S.K3(0,0):Yufx(x,y)Ynfy(x,y)3?Yfy(0,0)=03(0,0):?.qf fx(0,0)x fy(0,0)y=px2+y2sin1x

17、2+y2 0(0)Kf(x,y)3(0,0):.9.ep(1)u=xsin(x+y)+y cos(x+y)k(2)u=12ln(x2+y2)k(3)u=xln(xy)3ux2y(4)u=ln(ax+by+cz)4ux4,4ux2y2(5)u=(x x0)p(y y0)qp+quxpyq(6)u=x y zex+y+zp+q+ruxp yq zr)(1)ux=(1 y)sin(x+y)+xcos(x+y),uy=y sin(x+y)+(x+1)cos(x+y)Kux2=(2 y)cos(x+y)xsin(x+y),uxy=uyx=(1 y)cos(x+y)(x+1)sin(x+y),uy2=y

18、cos(x+y)(x+2)sin(x+y)(2)ux=xx2+y2,uy=yx2+y2Kux2=y2 x2(x2+y2)2,uxy=uyx=2xy(x2+y2)2,uy2=x2 y2(x2+y2)2(3)uy=xyK2uxy=1yu3ux2y=0(4)ux=aax+by+cz,uy=bax+by+cz2ux2=a2(ax+by+cz)2,2uy2=b2(ax+by+cz)23ux3=2a3(ax+by+cz)3,3uxy2=2ab2(ax+by+cz)34ux4=6a4(ax+by+cz)4,4ux2y2=6a2b2(ax+by+cz)4(5)quyq=q!(x x0)pKp+quxpyq=p

19、!q!(p,qg,)(6)p+q+ruxp yq zr=pxp(xex)qyq(yey)rzr(zez)=ex+y+z(x+p)(y+q)(z+r)10.213(1)u=x2 2xy 3y2(2)u=xy2(3)u=arccossxy?y2uxy=2uyxy(1)ux=2x 2y,uy=2x 6yK2uxy=2,2uyx=2u2uxy=2uyx(2)ux=y2xy21,uy=2yxy2lnxK2uxy=2yxy21(1+y2lnx),2uyx=2yxy21(1+y2lnx)u2uxy=2uyx(3)?0 x 6 yu=arccossxy=arccosxyKux=12px(y x),uy=x2y

20、y xu2uxy=14x(y x)32,2uyx=14x(y x)32l2uxy=2uyxny?y 6 x 0)uxzx+yzy=z=px2+y2.9.?yz=x yx!+yx!216vx22zx2+2xy2zxy+y22zy2=0yzx=yx!yx0 yx!yx20 yx!,zy=0 yx!+1x0 yx!K2zx2=y2x300+2yx30+y2x400,2zxy=yx2001x20yx300,2zy2=1x00+1x200ux22zx2+2xy2zxy+y22zy2=010.u=(x+at)+(x at),?gy2ut2=a22ux2.yu=(x+at)+(x at),?gKut=a(0

21、 0),ux=0+0u2ut2=a2(00+00),2ux2=00+00l2ut2=a22ux2.2173.d(|)(1.de(z=f(x,y)(1)x+y+z=ez(2)xyz=x+y+z)(1)ux1+zx=zxezKzx=1ez 1uzx2=ez(1 ez)3zy=1ez 1,zy2=ez(1 ez)3,zxy=zyx=ez(1 ez)3(2)uxyz+xyzx=1+zx()Kzx=yz 11 xy()ux2yzx+xyzx2=zx2Kzx2=2yzx1 xy=2y(yz 1)(xy 1)2zy=xz 11 xy,zy2=2x(xz 1)(xy 1)2,zxy=zyx=2z(xy 1)2

22、2.de(1)f(x+y,y+z,z+x)=0zx,zy(2)z=f(xz,z y)dz(3)F(x y,y z,z x)=0zx,zy(4)F(x,x+y,x+y+z)=0zx,zy,2zx2.)(1)uxz=z(x,y)f1+f2zx+f3(zx+1)=0Kzx=f1+f3f2+f3zy=f1+f2f2+f3(2)dz=(xdz+z dx)f1+(dz dy)f2Kdz=zf1dx f2dy1 xf1 f2(3)uxz=z(x,y)F1 F2zx+F3(zx 1)=0Kzx=F1 F3F2 F3zy=F2 F1F2 F3(4)uxz=z(x,y)F1+F2+F3(1+zx)=0()Kzx=

23、F1+F2+F3F33()2uxF11+F12+F13(1+zx)+F21+F22+F23(1+zx)+zx2F3+(1+zx)F13+F23+F33(1+zx)=0Kzx2=1F33F23(F11+2F12+F22)2F3(F1+F2)(F13+F23)+F33(F1+F2)2zy=F2+F3F33.dz=x+y (z)(z=z(x,y)1 y0(z)6=0yzy=(z)zxyz=z(x,y)dz=dx+(z)dy+y0(z)dzq1 y0(z)6=0Kdz=dx+(z)dy1 y0(z)uzy=(z)1 y0(z),zx=11 y0(z)lzy=(z)zx2184.ydax+by+cz=(

24、x2+y2+z2)z=z(x,y)v(cy bz)zx+(az cx)zy=bx ay(u)ua,b,c.yz=z(x,y)(u)uKadx+bdy+cdz=2(xdx+y dy+z dz)0uzx=2x0 ac 2z0,zy=2y0 bc 2z0l(cy bz)zx+(az cx)zy=bx ay5.?yd(cx az,cy bz)=0z=z(x,y)vazx+bzy=c.yOux,yz=z(x,y)c1 a1zx b2zx=0,a1zy+c2 b2zy=0uzx=c1a1+b2,zy=c2a1+b2lazx+bzy=c.6.ydF(x+zy1,y+zx1)=0(z=z(x,y)vxzx+y

25、zy=z xy.yOux,yz=z(x,y)F1 1+zxy!+F2 zxxzx2!=0,F1 zyyzy2!+F2 1+zyx!=0uzx=yzF2 x2yF1x(xF1+yF2),zy=xzF1 xy2F2y(xF1+yF2)lxzx+yzy=z xy.7.e|(1)?x+y+z=0,x y z=1,dydx,dzdx,d2ydx2(2)x+y=u+v,xy=sinusinv,du,dv(3)?xu+yv=0,yu+xv=1,ux,uy,vx,vy,2uxy(4)x=coscos,y=cossin,z=sin,zx,zy(5)?u=f(u,x,v+y),v=g(u x,u2 y),ux,v

26、x.)(1)x1+dydx+dzdx=0yz+xzdydx+xydzdx=0()dydx=y(z x)x(y z),dzdx=z(x y)x(y z)()2xd2ydx2+d2zdx2=0zdydx+ydzdx+zdydx+xdydxdzdx+xzd2ydx2+ydzdx+xdydxdzdx+xyd2zdx2=0d2ydx2=2zdydx+2ydzdx+2xdydxdzdxx(y z)dydx,dzdxd2ydx2=yz(x y)2+(x z)2+(y z)2x2(z y)3219(2)?U?u+v=x+yy sinu=xsinv?du+dv=dx+dysinudy+y cosudu=sinv

27、 dx+xcosv dvKdu=1xcosv+y cosu(sinv+xcosv)dx (sinu xcosv)dydv=1xcosv+y cosu(sinv y cosu)dx+(sinu+y cosu)dy(3)?xdu+y dv=udx v dyy du+xdv=v dx udyudu=1x2 y2(yv xu)dx+(yu xv)dy,dv=1x2 y2(yu xv)dx+(yv xu)dyKux=yv xux2 y2,uy=yu xvx2 y2,vx=yu xvx2 y2,ux=yv xux2 y2u2uxy=(yux v xvx)(x2 y2)2x(yu xv)(x2 y2)2ux

28、,vx2uxy=2(x2v+y2v 2xyu)(x2 y2)2(4)dx,yx1=sin cosx cos sinx0=sin sinx+cos cosxKx=cossin,x=sincosuzx=zx=cotcos=xznzy=yz(5)xux=f1ux+f2+f3vxvx=g1 ux 1!+2vyg2vxKux=f2(1 2vyg2)g1f3(f1 1)(2vyg2 1)g1f3,vx=g1(f1+f2 1)(f1 1)(2vyg2 1)g1f3.8.x=u+v,y=u2+v2,z=u3+v3zx,yzx,zy.)x2 y=2uvKz=(u+v)(u2 uv+v2)=x2(3y x2)uz

29、x=32(y x2),zy=32x.9.x=rcos,y=rsinCdxdt=y+kx(x2+y2)dydt=x+ky(x2+y2)4I.)dx,ytl4ICr,tx=rcos,y=rsintdxdt=cosdrdt rsinddtdydt=sindrdt+rcosddtx,y,dxdt,dydt?|cosdrdt rsinddt=rsin+krcos r2sindrdt+rcosddt=rcos+kr sin r2udrdt=kr3,ddt=1.10.x=eucos,y=eusin C2zx2+2zy2=0.)x=eucos,y=eusinKu=ln(x2+y2),=arctanyx220K

30、ux=xx2+y2,uy=yx2+y2;x=yx2+y2,y=xx2+y2uux=y,uy=xqzx=zuux+zx,zy=zuuy+zyK2zx2=2zu2 ux!2+22zuuxx+2z2 x!2+zu2ux2+z2x22zy2=2zu2 uy!2+22zuuyy+2z2 y!2+zu2uy2+z2y2q2ux2=x y!=y x!=y uy!=2uy22x2=2y2K2x2+2y2=2ux2+2uy2=0q ux!2+uy!2=x!2+y!2,uxx=uyyK2zx2+2zy2=e2u 2zu2+2z2!=0=2zu2+2z2=0.11.x=rcos,y=rsinKf(x,y)=(r,)

31、ur,5L2fx2+2fy2.)f(x,y)=(r,)ur,fxxr+fyyr=r=fxcos+fysin=rfxx+fyy=fxrsin+fyrcos=K2r2=2fx2cos2+2fxysin2+2fy2sin222=r2 2fx2sin2 2fxysin2+2fy2cos2!fxrcos fyrsinu2r2+1r222=2fx2+2fy21rrl2fx2+2fy2=2r2+1r222+1rr12.x=e,y=eCax22zx2+2bxy2zxy+cy22zy2=0(a,b,c).)x=e,y=eK=lnx,=lnyuddx=1x,ddy=1yKzx=1xz,zy=1yzu2zx2=1x

32、2 2z2z!,2zy2=1y2 2z2z!,2zxy=1xy2z?zna 2z2z!+2b2z+c 2z2z!=0.13.=x,=x2+y2Cyzx xzy=0.)dzx,y,qx,ylzwLmC,ux,yEuzx=z+2xz,zy=2yzyzx xzy=yz221y 6 0Kdyzx xzy=0z=0.14.=x,=y x,=z xCux+uy+uz=0.)dux,y,z,qx,y,zluwLmC,ux,y.zEuux=uuu,uy=u,uz=uKdux+uy+uz=0z=015.5C=x+1y,=x+2yy3rA2ux2+2B2uxy+C2uy2=0(A,B,CAC B2 0)?:3Iu

33、a.y3?:P0(x0,y0,z0)K3T:12x0(x x0)+12y0(y y0)+12z0(z z0)=0=y0z0(x x0)+x0z0(y y0)+x0y0(z z0)=0u3Iax0,ay0,az0a(x0+y0+z0)=a.4.x2+y2=a2,bz=xy?.)?:M0(x0,y0,z0)d3M0:n1=2x0,2y0,0,n2=y0,x0,bu?vcos=n1 n2|n1|n2|=2bz0|a|a2+b2.2266.F1.u=x2 xy+y23(1,1)?l=(cos,sin).?(1)3=k(2)3=k?(3)3=0(4)uF.)ux=2x y,uy=x+2yKux(1,1)

34、=1,uy(1,1)=1qul=ux(1,1)cos+uy(1,1)sinKul=cos+sin=2sin +4!u(1)?=43l=22,22!k2(2)?=343l=22,22!k?2(3)?=4,343l=22,22!l=22,22!0(4)gradu=ux(1,1)i+uy(1,1)j=i+j.2.u=xyz3:M(1,1,1)l=(2,1,3)9F.)ux=yz,uy=xz,uz=xyK3(1,1,1):ux=uy=uz=1qlucos=214,sin=114,cos=314Kul=ux(1,1,1)cos+uy(1,1,1)cos+uz(1,1,1)cos=2714 gradu=i

35、+j+k.3.u=x2+2y2+3z2+xy+3x 2y 6z3O(0,0,0)9A(1,1,1)F9?.)ux=2x+y+3,uy=4y+x 2,uz=6z 6K3O(0,0,0):ux=3,uy=2,uz=6ugradu=3i 2j+6k,|gradu|=73A(1,1,1):ux=6,uy=3,uz=0ugradu=6i+3j,|gradu|=35.4.y(1)grad(u+v)=gradu+gradv,(2)grad(uv)=ugradv+vgradu(3)gradF(u)=F0(u)graduy?5y.-u=u(x,y),v=v(x,y)(1)(u+v)x=ux+vx,(u+v)y=

36、uy+vyKgrad(u+v)=(u+v)x,(u+v)y!=ux,vy!+ux,vy!=gradu+gradv.(2)(uv)x=vux+uvx,(uv)y=vuy+uvyKgrad(uv)=(uv)x,(uv)y!=v ux,uy!+u vx,vy!=ugradv+vgradu.(3)gradF(u)=Fx,Fy!=F0(u)ux,F0(u)uy!=F0(u)ux,uy!=F0(u)gradu?y.5.ygrad1r=rr3r=px2+y2+z2,r=xi+yj+zk.yrx=xr,ry=yr,rz=zrKgrad1r=ddr 1r!gradr=1r2 rxi+ryj+rzk!=1r21r

37、(xi+yj+zk)=rr3.2276.u=ln1r,r=p(x a)2+(y b)2+(z c)23m=:|gradu|=1)rx=x ar,ry=y br,rz=z crKgradu=ddr ln1r!gradr=1r x ari+y brj+z crk!=1r2(x a)i+(y b)j+(z c)ku|gradu|=1r=1Kr=1=(x a)2+(y b)2+(z c)2=1L3(a,b,c)%1|gradu|=1.2287.V1.:(1,2)NCf(x,y)=2x2 xy y2 6x 3y+5V.)fx=4x y 6,fy=x 2y 3,fx2=4,fxy=1,fy2=2kn0K3

38、:(1,2)f=5,fx=0,fy=0,fx2=4,fxy=1,fy2=2uf(x,y)=5+2(x 1)2(x 1)(y+2)(y+2)2.2.Ux9ymf(x,y)=exln(1+y)n.)fx=exln(1+y),fy=ex1+y,fx2=exln(1+y),fy2=ex(1+y)2,fxy=ex1+yfx3=exln(1+y),fy3=2ex(1+y)3,fxy2=ex(1+y)2,fyx2=ex1+yK3:(0,0)?f=0,fx=fx2=fx3=0,fy=1,fxy=1,fy2=1,fxy2=1,fyx2=1,fy3=2uf(x,y)=y+xy 12y2+12x2y 12xy2+1

39、3y32291441.4?1.e4(1)z=x2(y 1)2(2)z=(x y+1)2(3)z=3axy x3 y3(a 0)(4)z=sinx+cosy+cos(x y)?x06y62?(5)z=xy r1 x2a2y2b2(a,b 0)(1)-zx=2x=0,zy=2(y 1)=0Kx=0,y=1u:(0,1)U4:qzx2=2,zxy=0,zy2=2KA=2,B=0,C=2uH=4 0Kz3x y+1=0:4?z=0.(3)-zx=3ay 3x2=0,zy=3ax 3y2=0K?x1=0y1=0?x2=ay2=a=3:(0,0),(a,a)?Uk4qzx2=6x,zxy=3a,zy2=6

40、yK3:(0,0)A=0,B=3a,C=0uH=9a2 0d43:(a,a)A=6a 0dk4z=a3.(4)-zx=cosx sin(x y)=0,zy=siny+sin(x y)=0Kcosx=sinyuy=2 xcosx sin(x y)=cos sin 2x 2!=2cosx2cos32x=00 6 x 62Kcosx26=0,cos32x=0ux=3,y=6=3:3,6!?Uk4qzx2=sinx cos(x y),zxy=cos(x y),zy2=cosy cos(x y)KA=3 0=3:3,6!?k4z=323.(5)-zx=yr1 x2a2y2b2x2ya2r1 x2a2y2

41、b2=0,zy=xr1 x2a2y2b2xy2b2r1 x2a2y2b2=0K?x1=0y1=0 x2=a3y2=b3x3=a3y3=b3x4=a3y4=b3x5=a3y5=b3u3:P1(0,0),P2 a3,b3!,P3 a3,b3!,P4 a3,b3!,P5 a3,b3!?U4qzx2=3a2b2xy+2b2x3y+3a2xy3a4b2?1 x2a2y2b2?32,zxy=a4b4 3a2b4x2+2b4x4 3a4b2y2+3a2b2x2y2+2a4y4a4b4?1 x2a2y2b2?32zy2=3b2x3y 3a2b2xy+2a2xy3a2b4?1 x2a2y2b2?322303:P

42、2 a3,b3!,P3 a3,b3!?A=43b3a 0uk4z=39ab3:P4 a3,b3!,P5 a3,b3!?A=43b3a 0,B=233,C=43a3bdH=4 0uk4?z=39ab3:P1(0,0)?A=0,B=1,C=0dH=1 0u4.2.yz=(1+ey)cosx yeyk4?4?.y-zx=(1+ey)sinx=0zy=eycosx ey yey=01+ey6=0Ksinx=0ux=k(k Z)qey6=0Kcosx 1 y=0=ky=cosx 1?ky=0?ky=2KU4:?x1=2ky1=0?x2=(2k+1)y2=2(k=0,1,2,)qzx2=(1+ey)cos

43、x,zxy=eysinx,zy2=eycosx 2ey yeyK3:(2k,0)A=2 0Kdk4z=23:(2k+1),2)A=1+1e2,B=0,C=1e2dH=1e2 1+1e2!Ox,y,zKC=x+y+z=2p D=x+y+z2=p uz=2p x yKS=pD(D x)(D y)(D z)=pp(p x)(p y)(x+y p)u=S2=p(p x)(p y)(x+y p),0 x,y pS4u4?u3:(x,y)40S3:(x,y)4-ux=p(p y)(2p 2x y)=0,uy=p(p x)(2p x 2y)=0p 6=0,0 x,y 0,B=4,C=18uH=2 0Kdk4

44、?z=5lk$:(1,1,5)q?x2+y2+z +p:.5.y=ax2+bx+cy|(xi,yi),i=1,2,n|?Xa,b,cvn?g|.)d=nXi=1(yi ax2i bxi c)2o(a,b,c)?d47ka=2nXi=1(yiax2ibxic)x2i=0,b=2nXi=1(yiax2ibxic)xi=0,c=2nXi=1(yiax2ibxic)=0=a,b,cven?g|nXi=1x4i!a+nXi=1x3i!b+nXi=1x2i!c=nXi=1x2iyi nXi=1x3i!a+nXi=1x2i!b+nXi=1xi!c=nXi=1xiyi nXi=1x2i!a+nXi=1xi!b

45、+nc=nXi=1yi6.y=x230,1o=ax+b5OUJ(a,b)=Z10(y )2dx4?eZCq231)J(a,b)=Z10(y )2dx=Z10(x2 ax b)2dx=15+a23+b2a223b+abJa,bJ(a,b)4?d47k-Ja=12+23a+b=0,Jb=23+a+2b=0Ka=1,b=16uy=x2=x 165OZCq.2322.41.e3e4(1)f=x+y,ex2+y2=1(2)f=x 2y+2z,ex2+y2+z2=1(3)f=xyz,e1x+1y+1z=1a(x 0,y 0,z 0,a 0)(4)f=1x+1y,ex+y=2(5)f=xyz,ex2+y2+

46、z2=1,x+y+z=0.)(1)L=x+y+(x2+y2 1)|Lx=1+2x=0Ly=1+2y=0L=x2+y2 1=0 x1=22y1=221=22x2=22y2=222=22qLx2=2,Lxy=0,Ly2=2Kd2L 22,22!=2(dx2+dy2)0u3 13,23,23!?4?3n3 13,23,23!?43.(3)L=xyz+1x+1y+1z1a!)|Lx=yz x2=0Ly=xz y2=0Lz=xy z2=0L=1x+1y+1z1a=0 x=y=z=3a,=81a4qLx2(3a,3a,3a)=Ly2(3a,3a,3a)=Lz2(3a,3a,3a)=6a,Lxy(3a,3a

47、,3a)=Lxz(3a,3a,3a)=Lyz(3a,3a,3a)=3aKd2L(3a,3a,3a)=3a(dx+dy+dz)2+dx2+dy2+dz2 0u3(3a,3a,3a)?4?27a3.(4)L=1x+1y+(x+y 2)233)|Lx=1x2+=0Ly=1y2+=0L=x+y 2=0 x=y=1qLx2(1,1)=Ly2(1,1)=2,Lxy(1,1)=0Kd2L(1,1)=2(dx2+dy2)0u3(1,1)?4?2.(5)L=xyz+u(x2+y2+z2 1)+v(x+y+z)|Lx=yz+2ux+v=0Ly=xz+2uy+v=0Lz=xy+2uz+v=0Lu=x2+y2+z2

48、1=0Lv=x+y+z=0 x1=66y1=66z1=63u1=612v1=16x2=66y2=66z2=63u2=612v2=16x3=63y3=66z3=66u3=612v3=16x4=63y4=66z4=66u4=612v4=16x5=66y5=63z5=66u5=612v5=16x6=66y6=63z6=66u6=612v6=16qd2L=2u(dx2+dy2+dz2)+2(z dxdy+y dxdz+xdy dz)K3:(x1,y1,z1)?d2L=66(dx2+dy2+dz2 4dxdy+2dxdz+2dy dz)dx2+y2+z2=12xdx+2y dy+2z dz=0K3:(x

49、1,y1,z1)?kdx+dy=2dzqdx+y+z=0dx+dy+dz=0Kdx=dy,dz=0ud2L(x1,y1,z1)=6 dx2 0K3 66,66,63!?4?618n3(x3,y3,z3),(x5,y5,z5)?4?6183(x2,y2,z2),(x4,y4,z4),(x6,y6,z6)?4618.2.f=xmynzp3x+y+z=a,a 0,m 0,n 0,p 0,x 0,y 0,z 0e.)x 0,y 0,z 0Kf=xmynzplnf=mlnx+nlny+plnz,Ilnf4:f4:-L=mlnx+nlny+plnz+(x+y+z a)K)Lx=mx+=0Ly=ny+=0L

50、z=pz+=0L=x+y+z a=0 x=mam+n+py=nam+n+pz=pam+n+p=m+n+paK mam+n+p,nam+n+p,pam+n+p!U4:qLx2=mx2,Lxy=Lyz=Lxz=0,Ly2=ny2,Lz2=pz2,d2L=mx2dx2ny2dy2pz2dz2!.?x+y=az=0?x+z=ay=0?y+z=ax=0f 0f4:.3.x2+3y2=12Sn/.1u.)duKn/Sux2(23)2+y24=1n/.1u234.:7:(0,2)n/,:I(x,y)(x,y 0)KSn/.2xpy+2n/n:IA(0,2),B(x,y),C(x,y)d5A(0,2),B(x