I have the following past exam question.

Write down the Cartesian coordinates x and y in terms of the plane polar coordinates r and

Which is easy.

Next, evaluate the jacobian determiant

By considering the matrix equation relating the vectors

and

use this result to obtain an expression for the area element dxdy in plane polar coordinates.

b)Work out the numerical value of the same area integral

over the interior of the circle of radius 2 centred at the origin.