UPSC 2021 Maths Optional Paper 1 Q3a-i — Step-by-Step Solution

7 marks · Section A

Jacobian · Calculus · asked 4× in 13 yrs · Read the full method →

Question

$u=x^2+y^2$ , $v=x^2-y^2$ , $x=r\cos\theta$ , $y=r\sin\theta$ . Find $\dfrac{\partial(u,v)}{\partial(r,\theta)}$ .

Express $u,v$ in $(r,\theta)$ , compute partials, take determinant.

$u=x^2+y^2=r^2$ .

$v=x^2-y^2=r^2\cos^2\theta-r^2\sin^2\theta=r^2\cos 2\theta$ .

$\partial u/\partial r=2r$ , $\partial u/\partial\theta=0$ .

$\partial v/\partial r=2r\cos 2\theta$ , $\partial v/\partial\theta=-2r^2\sin 2\theta$ .

\dfrac{\partial(u,v)}{\partial(r,\theta)}=\det\begin{pmatrix}\partial u/\partial r & \partial u/\partial\theta\\ \partial v/\partial r & \partial v/\partial\theta\end{pmatrix}=\det\begin{pmatrix}2r & 0\\ 2r\cos 2\theta & -2r^2\sin 2\theta\end{pmatrix}.

$=(2r)(-2r^2\sin 2\theta)-(0)(2r\cos 2\theta)=-4r^3\sin 2\theta$ .

\boxed{\;\dfrac{\partial(u,v)}{\partial(r,\theta)}=-4r^3\sin 2\theta.\;}