How to find the tangent space of a general submanifold?

Question

Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general submanifolds?

Answer 1

For a point $p \in \phi(S)$, put local coordinates $(x_1,\dots,x_n)$ on $S$ around the point $\phi^{-1}(p)$. Then the tangent space $T_{\phi^{-1}(p)}S$ is spanned by $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}$. The tangent space $T_p\phi(S)$ as a subset of $T_pM$ should then be the image of $T_{\phi^{-1}(p)}S$ under the pushforward map $\phi_*$. That is, the space you are looking for should be the linear span of $\phi_* \frac{\partial}{\partial x_1}, \dots, \phi_* \frac{\partial}{\partial x_n}$ in $T_pM$.