Solution
= Solution
{c}
A <tangent vector> to $M$ at $p$ is a velocity $\gamma'(0)$ of a smooth curve $\gamma:(-\varepsilon,\varepsilon)\to M$ with $\gamma(0)=p$. The <tangent space> is the set of all such velocities:
$$
\boxed{T_pM=\{\gamma'(0):\gamma(0)=p,\ \gamma\text{ a smooth curve in }M\}.}
$$
It is a $k$-dimensional linear subspace of $\mathbb R^n$; for a local parametrization $\phi$, it is the image of the injective <derivative> $D\phi$ at the corresponding parameter point.