Question

设z=f(x,u,v),u=e^xy,v=x^2-y^2,且f具有二阶连续的偏导数，求�6�8^2z/�6�8x�6�8y。要具体过程。

Answer 1

令e^xsiny=u,x^2+y^2=v
则δz/δx
=δf/δu*δu/δx+δf/δv*δv/δx
=δf/δu*(e^xsiny)+δf/δv*(2x)
δ^2z/δx^2
=δ^2f/δu^2*(e^xsiny)*(e^xsiny)+δ^2f/δuδv*(2x)*(e^xsiny)+δf/δu*(e^xsiny)+δ^2f/δvδu*(e^xsiny)*2x+δ^2f/δv^2*(2x)*(2x)+2δf/δv
=(e^2x*(siny)^2)*δ^2f/δu^2+(e^xsiny)* δf/δu+(4xe^xsiny)*δ^2f/δuδv+4x^2*δ^f/δ^2v+2δf/δv
(f（u，v）具有二阶连续偏导数=>δ^2f/δuδv=δ^2f/δvδu)