# Question on the definition of tangent space at boundary points

I'm going thought Spivak's 'Calculus On Manifolds' and I am bit confused with certain things in the definition of outward normal unit vector.

For the notation. With $$v_{p}$$ Spivak denotes the tangent vector $$v$$ at the point $$p$$, for a function $$f:A\to\mathbb{R}^m$$ with $$A\subseteq\mathbb{R}^n$$ differentiable on $$A$$ we have $$f_{\ast}(v_p):=(Df(p)(v))_{f(p)}$$ and for a basis $$v_1,\ldots,v_n$$ he denotes by $$[v_1,\ldots,v_n]$$ the orientation of $$v_1,\ldots,v_n$$.

Spivak defines coordinate systems for manifolds wich then he uses in the definition of tangent space. He does not define coordinate systems on manifolds with boundaries but, nevertheless, he uses them in the definition of outward normal unit vectors. Here I read how to extend the definition of coordinate systems to manifolds with boundary but I haven't been able to extend the definition of tangent space to points in the boundary. How do we do that? Also, why is the tangent space $$\partial M_{x}$$ a subspace of $$M_{x}$$?

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For manifolds with boundary, the coordinate condition should just be modified slightly, taking $$W$$ to be an open subset of $$H^k = \{(x_1,\dots,x_k)\in \mathbb{R}^k : x_k\ge 0\}$$, a half space in $$\mathbb{R}^k$$. Since $$\mathbb{R}^k$$ is diffeomorphic to the interior of $$H^k$$, you get the interior points of $$M$$ using $$W$$ contained in the interior of $$H^k$$, and you get the boundary points of $$M$$ by using $$W$$ which intersect the boundary of $$H^k$$. You might ask what it means for a function $$f:W \to \mathbb{R}^n$$ to be smooth if $$W \subset H^k$$ has boundary points. It simply means there is an open set $$V \subset \mathbb{R}^k$$ with $$V \cap H^k = W$$ and an extension $$f_V:V\to \mathbb{R}^n$$ which is smooth and which restricts to $$f$$ on $$W$$, i.e., $$f_V|_W = f$$. Then the tangent space is similar to before, taking the image of the derivative of the extension, i.e., if $$a\in \partial W$$ and $$f(a) = x \in \partial M$$, then $$M_x = (f_V)_*(\mathbb{R}^k_a)$$.
Now, let's keep the notation $$a\in \partial W$$ and $$f(a) = x \in \partial M$$. We have $$\partial W = W \cap \partial H^k$$, and let's define $$g:\partial W \to \mathbb{R}^n$$ by $$g=f|_{\partial W}$$. This is a local coordinate system for $$\partial M$$ at $$x$$, identifying $$\partial H^k$$ with $$\mathbb{R}^{k-1}$$ in the natural way. You'll see that the Jacobian of $$g$$ at $$a$$ is just the Jacobian of $$f_V$$ at $$a$$ without the $$k^{\small\text{th}}$$ column. That's one way you can see that $$(\partial M)_x = g_*(\mathbb{R}^{k-1}_a)$$ is a subspace of $$M_x$$, and there are less coordinate-reliant ways to say it, too, but this is pretty concrete.