Local Coordinate Bases#

Theorem: Local Coordinate Bases

Let \((U, \phi)\) be a chart on the Euclidean space \(\mathbb{R}^n\) and let \(\tau: \phi(U) \subseteq \mathbb{R}^n \to \mathbb{R}^n\) be the transition map from \((U, \phi)\) to Cartesian coordinates.

If \(\tau\) is totally differentiable at \(\phi(\boldsymbol{p})\) and the determinant of its Jacobian matrix \(J_{\tau}(\phi(\boldsymbol{p}))\) is non-zero, then the columns of \(J_{\tau}(\phi(\boldsymbol{p}))\) form a Hamel basis for \(\mathbb{R}^n\).

Proof

TODO