$u=x^2-y^2=r^2 (-\sin^2 \theta + \cos^2 \theta ) = r^2 \cos 2 \theta$ $u_\theta = -2r^2 \sin2 \theta$ $u_r= 2r \cos 2 \theta$ $v=2x^2-y^2 = r^2 (2 \cos^2 \theta - \sin^2 \theta ) = \dfrac {r^2 } {2} (1+3 \cos 2 \theta) $ $v_\theta = -3r^2 \sin 2 \theta$ $v_r = r (1+3 \cos 2 \theta)$ $\dfrac {\partial (u,v)}{\partial (r, \theta)} = \begin{vmatrix} \dfrac {\partial u}{\partial r} & \dfrac {\partial u}{\partial \theta} \ \dfrac {\partial v} {\partial r}&\dfrac {\partial v} {\partial \theta} \end{vmatrix} = \begin{vmatrix} u_r &u_\theta \ v_r &v_\theta \end{vmatrix} = u_rv_\theta - u_\theta v_r $ $=2r \cos 2\theta (-3r^2 \sin 2 \theta ) - ( -2 r^2 \sin 2 \theta ) r (1+3 \cos 2 \theta )$ $=-6r^3\sin 2 \theta \cos 2 \theta + 2r^2 \sin 2 \theta +6r^3 \sin 2 \theta \cos 2 \theta $ $ \dfrac {\partial (u,v)}{\partial (r,\theta)}= 2r^2 \sin 2 \theta$