<s>[INST] What is $df^{-1}(f(\begin{smallmatrix}x\\y\end{smallmatrix}))$? 
$$f:\mathbb{R}\times[0,2\pi) \to \mathbb{R}^2,f\begin{pmatrix}x \\y\end{pmatrix}=\begin{pmatrix}e^x \cos y \\e^x \sin y\end{pmatrix}$$

which is $\exp(z)$
I know that $df(x,y)=\begin{pmatrix}e^x\cos y && -e^x\sin y\\e^x\sin y&&e^x\cos y\end{pmatrix}$.
How can I determine $df^{-1}\Bigg(f\begin{pmatrix}x \\y\end{pmatrix}\Bigg)$?
 [/INST] We have $df^{-1}\Bigg(f\begin{pmatrix}x \\y\end{pmatrix}\Bigg)=(df(x,y))^{-1}=\begin{pmatrix}e^x\cos y && -e^x\sin y\\e^x\sin y&&e^x\cos y\end{pmatrix}^{-1}$.
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