# Showing that a certain function is a local diffeomorphism

I have to show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2: (x,y) \mapsto (e^x(x \cos y - y \sin y),e^x(x \sin y + y \cos y)$ is a local diffeomorphism in ever point not $(-1,0)$.

I have no idea how to invert $f$ on some restricted domain. My guess is that we have to restrict the domain, jut say cos and sin have inverses. However, I have no idea what I could pick as a viable inverse? Since taking logarithms, dividing the result and multiplying or squaring just makes the result more complicated. Could anyone give me a hint?

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## 1 Answer

Hint: Calculate Jacobian matrix for mapping $f: (x,y) \mapsto (e^x(x \cos y - y \sin y),e^x(x \sin y + y \cos y)$ and check its determinant is zero or not.

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Well if the jacobian is not 0 in a point, than the inverse function theorem tells us that it is a local diffeomorphism in that point. But the inverse function theorem is mentioned in the next section, not in this section. – Nga Nov 18 '12 at 18:46