Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 25J / a / Solution

A subset $X\subset {\mathbb{R}}^N$ is a smooth $k$-manifold if every $x\in X$ has a neighborhood parametrized by a smooth map $\varphi :U\subset {\mathbb{R}}^k\to X$ that is a homeomorphism onto its image and has derivative of rank $k$. DefineIf $\psi$ is another parametrization, the transition maps satisfyThe transition derivative is invertible, so the two images coincide. Thus the tangent space is independent of the parametrization.