⑩天掌 Teaching Plan on Advanced Mathematics o 定理2(充分条件)若函数z=(x,)的偏导数02,02 ax a y 在点(x,y)连续,则函数在该点可微分 证:△z=f(x+△x,y+△y)-f(x,y) =[f(x+△x,y+△y)-f(x,y+△y)]+|∫(x,y+△y)-f(x,y) =f(x+6△x,y+△y)△x+f(x,y+2△y)△y =U(x,y)& Ax+If,(x, y)+B Ay(0<01, 02<1) lim a=0, lim B=0 △x-0 △x→0 △y→>0 tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics = [ f (x + x, y + y) ] 定理2 (充分条件) y z x z , 证： z = f (x + x, y + y)− f (x, y) (0 , 1 ) = [ f x (x, y)+ ]x 1 2 f x y y y = f x (x +1 x, y + y)x + y ( , + 2 ) − f (x, y + y) +[ f (x, y + y ) − f (x, y)] f x y y +[ y ( , ) + ] 若函数 的偏导数 在点(x, y) 连续, 则函数在该点可微分. lim 0 0 0 = → → y x lim 0, 0 0 = → → y x
⑩天掌 Teaching Plan on Advanced Mathematics o △z=…=∫1(x,y)Ax+f1(x,y)△y+a△x+BA lim a=0, lim B=0 △x→>0 △x→0 △y→>0 J→>0 注意到△x+B△ax+B/,故有 Az=f(x,y)△x+f(x,y)△y+0(p) 所以函数z=f(x,y)在点(x,y)可微 tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics z = f x y x f x y y = x ( , ) + y ( , ) z f x y x f x y y = x ( , ) + y ( , ) + x + y 所以函数 + x + y 在点 可微. lim 0 0 0 = → → y x lim 0, 0 0 = → → y x 注意到 , 故有 + o( )