= 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 $\phi:U\subset\mathbb R^k\to X$ that is a homeomorphism onto its image and has <derivative> of rank $k$. Define
$$
T_xX=\operatorname{im}D\phi_u,\qquad \phi(u)=x.
$$
If $\psi$ is another parametrization, the transition maps satisfy
$$
D\phi_u=D\psi_vD(\psi^{-1}\circ\phi)_u.
$$
The transition <derivative> is invertible, so the two images coincide. Thus the tangent space is independent of the parametrization.
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