曲面的法向量与切线方程.ppt

主要内容平面点集和区域多元函数的极限多元函数连续的概念极限运算多元连续函数的性质多元函数概念全微分的应用高阶偏导数隐函数求导法则复合函数求导法则全微分形式的不变性方向导数全微分概念偏导数概念微分法在几何上的应用多元函数的极值练****题 ..,, ),,( 22 22 2yx zy zx zy xyfz????????求设 4.)(., ),,( 22 23 二阶偏导数连续求设 fyx zy zx y xyfxz?????? 2..)( ),( 2 2 2222x zfyxyxfz?????,求具有二阶导数设)( 二阶偏导数连续 f 5.., 2yx zezyx z??????求设 6..,),( ),,( ),,( 2x vx ugfyvxugv yv ux fu???????????,求偏导数连续设. lim )2(, )( lim )1( 220 0xy yxyx xxy y xy x?????????求极限 )0(, sin , cos )1(????????)0,0(),(???等价于则yx????? cos ) cos (sin )(0 222?????yx xxy cos ) cos (sin ??????,2??.0 )( lim 220 0?????yx xxy y x故. lim )2(, )( lim )1( 220 0xy yxyx xxy y xy x?????????求极限 xy yx y x????? lim )2(????????????xy y x11 lim .0? 2..)( ),( 2 2 2222x zfyxyxfz?????,求具有二阶导数设解令, 2222yxyxu???).(ufz?则 zu xyz 型 x udu dfx z??????).22( 2xxy f?????? fxxy xx z???????)22( 22 2 u xyz 型f ??? x fxxy xxy x f????????????)22()22( 22?????????????????x udu fdxxy yf)22()22( 22 )]22([)22()22( 222xyxfxxy yf???????????.)1(4)1(2 222fyxfy ??????? 3..,, ),,( 22 22 2yx zy zx zy xyfz????????求设)( 二阶偏导数连续 f 解令,xyu?;yv?).,(vufz?则记, 1u ff????, 12 2 11u fu ff?????????, 2v ff????v fvu ff?????????? 1 212, 12 222v fv ff?????????. 2 221u fuv ff??????????二阶偏导连续 x uu fx z????????, 1fy ??? z uv xy 型 v fy uu fy z???????????, 21ffx ????? x uu fx z????????, 1fy ???v fy uu fy z???????????, 21ffx ????? 1f ? uv xy 型 2f ? uv xy 型)( 12 2fyxx z??????x fy???? 1x uu fy???????? 1. 11 2fy ???)( 212 2ffxyy z????????y ffxy???????? 21)(y fy fx????????? 21??????????????????????????????????v fy uu fv fy uu fx 2211???? 22 21 12 11ffxffxx ????????????.2 22 12 11 2ffxfx ?????????)( 1 2fyyyx z???????y fyf?????? 11???????????????????v fy uu fyf 111 ).( 12 11 1ffxyf ???????? 4.)(., ),,( 22 23 二阶偏导数连续求设 fyx zy zx y xyfxz??????解令,xyu?;x yv?).,( 3vufxz?则记, 1u ff????, 12 2 11u fu ff?????????, 2v ff????v fvu ff?????????? 1 212, 12 222v fv ff?????????. 2 221u fuv ff??????????二阶偏导连续?? yvufxy z 3),(????y fx??? 3)( 3y vv fy uu fx??????????) 1( 21 3x fxfx??????. 2 21 4fxfx ???? f uv xy 型?? yfxfxy z 2 21 42 2