Let us say I started with the matrix $$ A= \begin{pmatrix} x & -y\\ y & x \\ \end{pmatrix} $$ And I wanted to use the tensor transformation: $$ \bar A^i_j=A^l_m \frac{\partial x_i'}{\partial x_l} \frac{\partial x_m}{\partial x'_j}$$ to transform to plane polar coordinates. What result would I expect? Here are my two thoughts:
- You are simply replacing $x$ with $r\cos(\theta)$ and $y$ with $r\sin(\theta)$ to get: $$ \bar A= \begin{pmatrix} r\cos(\theta) & -r\cos(\theta)\\ r\sin(\theta) & r\sin(\theta) \\ \end{pmatrix} $$
- Or we are performing a change of basis, in which case I think we should get: $$ \bar A= \begin{pmatrix} r & 0\\ 0 & -r \\ \end{pmatrix} $$ or something along those lines.
(Please note that I am not asking for a solution to the given problem, I want to know what the this transformation is doing, i.e. is it changing basis etc.)