Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 26I / a / Solution

For an embedded submanifold $X\subseteq {\mathbb{R}}^N$ and $p\in X$, the tangent space isEquivalently, in a local parametrization $\phi :U\subseteq {\mathbb{R}}^d\to X$ with $\phi (u)=p$, one has $T_{p}X=\mathrm{im}D{\phi }_u$.