曲面的法向量与切线方程.ppt

主要内容平面点集多元函数概念和区域多元函数极限运的极限多元连续函多元函数教的性质连块的概念[[全微分全微分方向号数概念的应用复合函数高阶偏号数求导法则偏导数概念隐函数全微分形式求导法则的不变性微分法在多元函数几何上的应用的极值[[(1)lim-r)r(2)limt+yx→0y-ovr+yx→∞xyy→>∞=∫(x2y2+x2+y2),(f具有二阶导数,=f(x,y,求ax2创2arby·(二阶偏导数连续)=x3f(xy,)求2,B→·(二阶偏导数连续)+y+x=2,=∫(ux,}+y),v=g(ux,v-y)(f,g偏导数连续,求a求极限(1)limr)rx→0以2,(2)linx+yy→0y→∞解(1)令x=pCos0,y=psin6,(p>0)则(x,y)→(0,0)等价于p→00v/=p(******@0)cos/=p(sin-c0s6cosb≤2p故linr)r0rtyx+y=linx->∞xyx→o(y→0y→>[[2设z=f(x2y2+x2+y2,(厂具有二阶导数,求a解令则z=f(u)azdfaux-“y型axduax=f(2ry"+2r)(2xy2+2x)f0(2xy2+2x)(2xy+2xaff(2y2+2)+(2xy2+2xdyf’anduaf'·(2y2+2)+(2xy2+2x)[f".(2x2y+2x)2(y2+1f"+4x2(y2+1)f".[[=∫(x,y,求97,97ax2'oy2aO(二阶偏导数连续)解令u=x,=y;则z=f(u,y).L一x记fm,=,a,/2sfdfi∫_fl二阶偏aoo导连续f21-avaazafauayauayav[[azafauaxauay·f1,azafauaff1+f2∫2<y型(y1)afia25(1+f2)=Q(对++:%1+,af2=x(x+邝1)+(xri1+f2)=x2f+2xf12+f2arayayy1)=f1+f1+升1+y(fil+f[[设z=x3f(xy,),求03,a(∫二阶偏导数连续)解令u=xy,p则z=x3f(au,v)记fif2f∫<×,型filLG。02f1二阶偏auoO导连续azf(,)afau,afLx3(f·x+f2)=x4f+x2f[[az=x'f(u,)y=+3af=r(ofou,afavx3(fx+f21)=x4f+x2/1(f+x22,2=()+(2ray+x292dfiau,aiaf2au,af2auayOvay=x4(f"·x+邝2·1)+x2(fx+/2x5邝+2x3f2+x/2,f12=f21axayayardr(xfi+rjf%)[[ar(xfi+xf2)(x)+x(x2)yf2+afiau,aiOn十x2xf2+xazauafzav4x3f+x4f1·y-f「2xf2+x2f21·y-f2[[