# 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 \(f^{-1}_{|f(U)} \in \mathcal{C}^1 (f(U))\)