Question

设z=f（xy,x+y）,且f有连续的二阶偏导数,求a^2z/axay

Answer 1

令u=xy,v=x+y

z=f(u,v)

az/ax=y(fu)+(fv)

a^2z/axay

=a(az/ax)/ay

=a(y(fu)+(fv))/ay

=(fu)+y(a(fu)/ay)+a(fv)/ay

=(fu)+y(a(fu)/au*au/ay+(a(fu)/av)*(av/ay))+((a(fv)/au)*(au/ay)+(a(fv)/av)*(av/ay))

=(fu)+(xy)*(fuu)+(x+y)*(fuv)+(fuv)