Question

$if \ u=r^2 \cos2\theta,v=r^2 \cos2\theta, \\ find, \dfrac{\partial (u,v)}{\partial (r,\theta )}$

$u=r^2 cos2\theta, v=r^2 sin2\theta \\ \dfrac{\partial u}{\partial r}=2r\cos2\theta, \dfrac{\partial u}{\partial \theta}=-2r^2\sin2\theta\\ \dfrac{\partial v}{\partial r}=2r\sin2\theta, \dfrac{\partial v}{\partial \theta}=2r^2\cos2\theta\\ $

$\dfrac{\partial (u,v)}{\partial (r,\theta )} =\begin{bmatrix}\dfrac{\partial u}{\partial r }& \dfrac{\partial v}{\partial \theta}\\\dfrac{\partial u}{\partial r}&\dfrac{\partial v}{\partial \theta}\end{bmatrix}\\ $

$=\begin{bmatrix}2r cos2\theta &-2 r^2 sin 2\theta\\ 2r sin2\theta &2r^2 cos 2\theta \end{bmatrix} $

$4r^3 cos^2 2\theta +4r^3 sin^2 2\theta\\ = 4r^3 (cos^2 2\theta + sin^2 2\theta)\\ =4r^3$$