Suppose $S\subset \Bbb R^n$ is a $k$-submanifold, is there a chart-independent (or "natural") way to identify $T_xS$ to a subspace of $\Bbb R^n$ for each $x\in S$?
If we let $\iota: S\to \Bbb R^n$ be the inclusion map, then I think we can naturally identify $T_xS$ with $d(\iota)_xT_xS$, a subspace of $\Bbb R^n$. Thing is, how to determine $d(\iota)_xT_xS$ chart-independently?
A chart-dependent way is to impose a chart $(\phi,U)$ about $x$, and $T_xS$ is spanned by $$\hat\partial_i=\left(C_x^\infty(S)\ni f\mapsto \frac\partial{\partial x_i}\Big|_{\phi(x)}f\circ\phi^{-1}\in\Bbb R\right),\quad i=1,\cdots,k$$ whose image under $d(\iota)_x$ is \begin{align} d(\iota)_x(\hat\partial_i) &=\left(C^\infty_x(\Bbb R^n)\ni g\mapsto \hat\partial_i \left(g\circ \iota\right)\in\Bbb R\right)\\ &= \left(C^\infty_x(\Bbb R^n)\ni g\mapsto \frac\partial{\partial x_i}\Big|_{\phi(x)} \left(g\circ \iota\circ\phi^{-1}\right)\in\Bbb R\right)\\ &=\left(C^\infty_x(\Bbb R^n)\ni g\mapsto \frac\partial{\partial x_i}\Big|_{\phi(x)} \left(g\circ\phi^{-1}\right)\in\Bbb R\right)\\ \end{align} And I get stuck at:
1). It looks quite hard to find a neat coordinate representation $d(\iota)_x(\hat\partial_i)=x^i\partial_i\in T_x\Bbb R^n=\Bbb R^n$ where $\partial_i$ is the standard partial derivative in the $i$-th component.
2). The dependence on $\phi$ seems unable to be unravelled.
So, I wonder: is there a way to determine the coordinate representation of $d(\iota)_xT_xS$ in $\Bbb R^n$ only via $x$ and not depending on $\phi$?
Update Actually from $$d(\iota)_x(\hat\partial_i) = \left(C^\infty_x(\Bbb R^n)\ni g\mapsto \frac\partial{\partial x_i}\Big|_{\phi(x)} \left(g\circ \iota\circ\phi^{-1}\right)\in\Bbb R\right)$$ I could have obtained $d(\iota)_x(\hat\partial_i) =\sum_{j=1}^n \dfrac{\partial}{\partial x_i}\Big|_{\phi(x)}(\iota\circ\phi^{-1})^j\partial_j$. And furthermore, we can see that the basis of $d(\iota)_xT_xS$ (identified in $\Bbb R^n$) is obtained by stacking these together as the columns of $D(\iota\circ\phi^{-1})|_{\phi(x)}$ where $D$ denotes the standard Jacobian matrix.
So my first question is in a sense solved. The remaining part is to undo the inpendence upon $\phi$.
Update 2 To undo the dependence on chart, we have only to show that given any two chart $\phi,\psi$ about $x$, $D(\iota\circ\phi^{-1})|_{\phi(x)}$ and $D(\iota\circ\psi^{-1})|_{\psi(x)}$ has the same column space. This observation may help solve the problem.
Update 3 (Problem totally solved) Okay guys I think I've been on the right track. The final observation is that $$D(\iota\circ\psi^{-1})|_{\psi(x)}=D(\iota\circ\phi^{-1})|_{\phi(x)}D(\phi\circ\psi^{-1})|_{\psi(x)}$$ in which $D(\phi\circ\psi^{-1})|_{\psi(x)}$ is clearly invertible since $\phi\circ\psi^{-1}$ is a diffeomorphism, this is sufficient to make the two indicated column spaces coincide. So my problem has been totally solved! But still many thanks to you guys who've been paying attention.
A chart independent definition of $T_x S$ is the set of all vectors $v \in \mathbb{R}^n$ for which there exists a smooth curve $\gamma : (-1,+1) \to \mathbb{R}^n$ satisfying the following conditions: