Question

If g(x, y) =f(u, v) where u=x^2-y^2 and v=2xy then prove that

Answer 1

Now, squaring and adding equations (i) and (ii),

u2+v2=x4−2x2y2+y4+4x2y2=x4+2x2y2+y4=(x2+y2)21∴x2+y2=u2+v2−−−−−−√u2+v2=x4−2x2y2+y4+4x2y2=x4+2x2y2+y4=(x2+y2)21∴x2+y2=u2+v2

Substituting this value in equation (vii),

(∂z∂x)2+(∂z∂y)2=4u2+v2−−−−−−√[(∂z∂u)2+(∂z∂v)2](∂z∂x)2+(∂z∂y)2=4u2+v2[(∂z∂u)2+(∂z∂v)2]

Hence Proved