# Local diffeomorphism

Calculus Level 5

Let $$f : X = \{(\rho,\phi,\theta) \in \mathbb{R}^3 \text{ ; } \rho > 0 \} \longrightarrow \mathbb{R}^3$$ defined as $f(\rho,\phi,\theta) = (\rho \cos \phi \sin \theta,\rho \sin \phi \sin \theta, \rho \cos \theta)$ Find where $$f$$ is a local diffeomorphism

Assumption.-

$$f$$ is a local diffeomorphism at a point $$x \in X \subseteq \mathbb{R}^3$$ if there exists a neighborhood $$U$$ of $$x$$ such that $$f_{|U}$$ has an inverse function $$f^{-1}_{|f(U)} \in \mathcal{C}^1 (f(U))$$

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