Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / F / 1 / a

A $k$-dimensional embedded submanifold of ${\mathbb{R}}^n$ is a subset $M$ such that near each $p\in M$ there is a smooth diffeomorphism of ambient open neighbourhoods that sends $M$ to the coordinate plane ${\mathbb{R}}^k\times \{0{\}}^{n-k}$. More precisely, for some open $U\ni p$ and diffeomorphism $\Phi :U\to V\subseteq {\mathbb{R}}^n$,Thus locally the subset has $k$ freely varying smooth coordinates.

A tangent vector to $M$ at $p$ is a velocity ${\gamma }^{\prime }(0)$ of a smooth curve $\gamma :(-\epsilon ,\epsilon )\to M$ with $\gamma (0)=p$. The tangent space is the set of all such velocities:It is a $k$-dimensional linear subspace of ${\mathbb{R}}^n$; for a local parametrization $\varphi$, it is the image of the injective derivative $D\varphi$ at the corresponding parameter point.