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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