Past exam of the short options course of the University of Oxford / 2023 / 4 / b / i / Solution

The curve-based tangent space at $x$ consists of velocities ${\gamma }^{\prime }(0)$ of smooth curves $\gamma :(-\epsilon ,\epsilon )\to M$ satisfying $\gamma (0)=x$. For an arbitrary subset these velocities need not form a vector space; for a smooth submanifold they do, and this is its usual tangent space. This definition suffices for the direct computation at the identity.