I read in a book: Consider the map $f:\mathbb{R}^2\to\mathbb{R}^2$ with $f(r,\theta)=(r',\theta')=(r^m,k\theta)$, where $k$ and $m$ natural numbers. The differential of this map sends the orthogonal frame $\frac{\partial}{\partial r}$ , $\frac{1}{r}\frac{\partial}{\partial \theta}$ to the frame $mr^{m-1}\frac{\partial}{\partial r'}$ and $kr^{m-1}\frac{1}{r'}\frac{\partial}{\partial \theta'}$.
My guess about what the differential(or the total derivative) looks like is that is the map induced by the matrix $\pmatrix{mr^{m-1}&0\\0&k/r}$ but probably that is not expressed in the correct base.
Can someone explain the correct way to do this?
By what the book says the matrix of the differential should look like $\pmatrix{mr^{m-1}& 0\\0 & kr^{m-1}}$.
The differential of $f$ in the polar coordinate basis $\{\partial_r, \partial_\theta\}$ is obtained by calculating the partial derivatives: $$ J_{\{\partial_r, \partial_\theta\}\mapsto \{\partial_r, \partial_\theta\}}(r, \theta) = \begin{pmatrix} mr^{m - 1}& 0 \\ 0 & k \end{pmatrix}. $$ If we want to express it relative to the basis $B = \{\partial_r, r^{-1}\partial_\theta\}$ in the domain and $B' = \{\partial_{r}, (r')^{-1}\partial_{\theta}\}$ in the codomain, we need a different matrix. Note that we maintain orthogonality and don't change the first basis vector so the only entry that will change is the bottom right entry. Also, we have: $$ df_{(r, \theta)}(r^{-1}\partial_{\theta}) = r^{-1}df_{(r, \theta)}(\partial_\theta) = r^{-1}k\partial_\theta = r^{-1}r' \frac{1}{r'}\partial_\theta = r^{m - 1}\cdot \frac{1}{r'}\partial_\theta. $$ So, this takes the second basis element of $B_1$ to $r^{m - 1}$ times the second basis element of $B_2$. This is what you wrote in your first paragraph. Thus the matrix with respect to these bases is $$ J_{B_1 \mapsto B_2}(r, \theta) = \begin{pmatrix} mr^{m - 1} & 0 \\ 0 & kr^{m - 1} \end{pmatrix}. $$ I believe the difference between your proposed matrix and the one the book gives is that you are using the same basis for the domain and the codomain, whereas the one in the book uses $\{\partial_r, r^{-1}\partial_\theta\}$ in the domain and $\{\partial_r, (r')^{-1}\partial_{\theta}\}$ in the codomain.