I see two ways of obtaining the desired result: either as in this answer (although not exactly the same) with a linear system (matrices) as in here:
$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \langle \mathrm{d} x, \frac{\partial}{\partial x} \rangle & \langle \mathrm{d} y, \frac{\partial}{\partial x} \rangle \\ \langle \mathrm{d} x, \frac{\partial}{\partial y} \rangle & \langle \mathrm{d} y, \frac{\partial}{\partial y} \rangle \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin \theta \\ -r \sin\theta & r \cos \theta \end{pmatrix} \cdot \begin{pmatrix} \langle \mathrm{d} r, \frac{\partial}{\partial r} \rangle & \langle \mathrm{d} \theta, \frac{\partial}{\partial r} \rangle \\ \langle \mathrm{d} r, \frac{\partial}{\partial \theta} \rangle & \langle \mathrm{d} \theta, \frac{\partial}{\partial \theta} \rangle \end{pmatrix} \cdot \begin{pmatrix} a_{x r} & a_{x \theta} \\ a_{y r} & a_{y \theta} \end{pmatrix} $$
The question is what is the purpose or meaning of the middle matrix in the very last part of the above expression (the scalar products of basis covectors and basis vectors)? Shouldn't it be the the identity matrix by definition (Kronecker delta)?
