Solución

El Jacobiano es

J(r,v)=(x,y)(x,v)=yuyvyryθ=2uv2vu=2u2+2v2J(r,v) = \frac{\partial(x,y)}{\partial(x,v)} = \begin{vmatrix} \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\\ \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix} = \begin{vmatrix} 2u & v\\ -2v & u \end{vmatrix} = 2u^2 + 2v^2