已知f(x,y)=x^2arctan(y/x)-y^2arctan(x/y)求它的混合二阶偏导

Question

已知f(x,y)=x^2arctan(y/x)-y^2arctan(x/y)求它的混合二阶偏导

Answer 1

f(x,y)=x^2arctan(y/x)-y^2arctan(x/y)
df(x,y)/dx=2xarctan(y/x)+x^2*1/(1+(y/x)^2)*(-y/x^2) -y^2*1/(1+(x/y)^2)*1/y
=2xarctan(y/x)-x^2y/(x^2+y^2)-y^3/(x^2+y^2)
=2xarctan(y/x)-(x^2y+y^3)/(x^2+y^2)
d^2f(x,y)/dxdy=2x*1/(1+(y/x)^2)*1/x -[(x^2+3y^2)(x^2+y^2)-(x^2y+y^3)*2y]/(x^2+y^2)^2
=2x^2/(x^2+y^2)-1
=(2x^2-x^2-y^2)/(x^2+y^2)
=(x^2-y^2)/(x^2+y^2)