Question

请问这道多元函数微分题怎么做？

如题,求大佬解答

Answer 1

Answer 2

已知g(x，y)=xy-f(x+y，x-y)，求∂²g/∂x²+∂²g/∂x∂y+∂²g/∂y²=?
解：设x+y=u，x-y=v；
则 ∂g/∂x=y-[(∂f/∂u)(∂u/∂x)+(∂f/∂v)(∂v/∂x)=y-∂f/∂u-∂f/∂v；
∂g/∂y=x-[(∂f/∂u)(∂u/∂y)+(∂f/∂v)(∂v/∂y)]=x-∂f/∂u+∂f/∂v;
∂²g/∂x²=-(∂²f/∂u²)(∂u/∂x)-(∂²f/∂v²)(∂v/∂x)=-∂²f/∂u²-∂²f/∂v²;
∂²g/∂y²=-[(∂²f/∂u²)(∂u/∂y)+(∂²f/∂v²)(∂v/∂y)]=-∂²f/∂u²+∂²f/∂v²;
∂²g/∂x∂y=(∂/∂y)(∂g/∂x)=1-(∂²f/∂u²)(∂u∂y)-(∂²f/∂v²)(∂v/∂y)=1-∂²f/∂u²+∂²f/∂v²;
∴∂²g/∂x²+∂²g/∂x∂y+∂²g/∂y²=(-∂²f/∂u²-∂²f/∂v²)+(1-∂²f/∂u²+∂²f/∂v²)+(-∂²f/∂u²+∂²f/∂v²)
=1-3(∂²f/∂u²)+∂²f/∂v²;