设f（u，v）具有二阶连续偏导数，z=f(e^xsiny，x^2+y^2). 计算δ^2z/δx^2 （δ为偏导数符号） 急求解答步

Question

如题所述

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)

Answer 2

过程如下:dz/dx=(df/du*du+df/dv*dv)/dx=f'1*(du/dx)+f'2*(dv/dx)=f'1*e^xsiny+f'2*2x,则d^2z/dx^2=(f''11*e^xsiny+f''12*2x)*e^xsiny+f'1*e^xsiny+(f''21*e^xsiny+f''22*2x)*2x+f'2*2(你自己整理一下,这里面有f''12=f''21)