Question

What is the Jacobian matrix J(r, θ) for the polar coordinate transformation, given that x=rcos⁡θ and y=rsin⁡θ.

Select one:

A. cos⁡θ-rsin⁡θ-sin⁡θrcos⁡θ

B. -rcos⁡θsin⁡θsin⁡θ-rcos⁡θ

C. cos⁡θ-rsin⁡θsin⁡θ-rcos⁡θ

D. cos⁡θrsin⁡θsin⁡θrcos⁡θ

Answer 1

$J=\left[ \begin{matrix} \frac{\partial}{\partial r}\left( r\cos \theta \right)& \frac{\partial}{\partial \theta}\left( r\cos \theta \right)\\ \frac{\partial}{\partial r}\left( r\sin \theta \right)& \frac{\partial}{\partial \theta}\left( r\sin \theta \right)\\\end{matrix} \right] =\left[ \begin{matrix} \cos \theta& -r\sin \theta\\ \sin \theta& r\cos \theta\\\end{matrix} \right] \\$

Only one '-' sign, no correct variant