Question

多元函数z=f(u,v),u=x+y,v=xy的二阶偏导数连续则f1'2'=f2'1'，该怎么理解

f1'f2是指对u,v求偏导还是对x,y分别求偏导？

Answer 1

对方程 z = f(y/x,x+2y) 的两端求微分,得 dz = f1*[(xdy-ydx)/x²]+f2*(dx+2dy) = [-(y/x²)f1+f2]dx+[(1/x)f1+2*f2]dy, 得到 Dz/Dx = -(y/x²)f1+f2,Dz/Dy = (1/x)f1+2*f2, 于是 D²z/DxDy = (D/Dx)(Dz/Dy) = (D/Dx)[(1/x)f1+2.