I have a couple questions on how Spivak defines an outward unit normal and orientation for the boundary of a manifold. I've included the relevant section of Calculus on Manifolds below. I should mention that when Spivak writes "manifold" they are referring to an embedded submanifold in $\mathbb{R}^n$. Spivak also uses the notation $H^k=\{x\in\mathbb{R}^k:x^k\geq 0\}$.
If $M$ is a $k$-dimensional manifold-with-boundary and $x\in\partial M$, then $(\partial M)_x$ is a $(k-1)$-dimensional subspace of the $k$-dimensional vector space $M_x$. Thus there are exactly two unit vectors in $M_x$ which are perpendicular to $(\partial M)_x$; they can be distinguished as follows. If $f\colon W\to \mathbb{R}^n$ is a coordinate system with $W\subset H^k$ and $f(0)=x$, then only one of these unit vectors is $f_*(v_0)$ for some $v_0$ with $v^k<0$. This unit vector is called the outward unit normal $n(x)$; it is not hard to check that this definition does not depend on the coordinate system $f$.
Suppose that $\mu$ is an orientation of a $k$-dimensional manifold-with-boundary $M$. If $x\in \partial M$, choose $v_1,\ldots,v_{k-1}\in (\partial M)_x$ so that $[n(x),v_1,\ldots,v_{k-1}]=\mu_x$. If it is also true that $[n(x),w_1,\ldots,w_{k-1}]=\mu_x$, then both $[v_1,\ldots,v_{k-1}]$ and $[w_1,\ldots,w_{k-1}]$ are the same orientation for $(\partial M)_x$. This orientation is denoted $(\partial\mu)_x$. It is easy to see that the orientations $(\partial \mu)_x$, for $x\in \partial M$, are consistent on $\partial M$.
Here are my questions:
How do we show that our definition of $n(x)$ is independent of our choice of coordinate system? Suppose $f\colon W\to\mathbb{R}^n$ and $g\colon V\to\mathbb{R}^n$ are two different coordinate systems for $M$ around a point $x\in \partial M$, where $W,V\subset {H}^k$ are relatively open sets and $f(0)=g(0)=x$. Then we have $$f_*(\mathbb{R}_0^{k-1}\times\{0\})=g_*(\mathbb{R}^{k-1}_0\times\{0\})=(\partial M)_x.$$ (Pardon the slight abuse of notation.) It would be nice to instead have $f_*({H}_0^k)=g_*({H}_0^k)$, but I'm not sure how to show this.
How do we show $(\partial \mu)_x$ is consistent? My guess is that we need to extend an arbitrary coordinate system for $\partial M$ into a coordinate system for $M$, and then make use of the fact that $\mu_x$ is consistent. However, I'm not sure how to approach either of these steps.
The first point comes because $\phi=f^{-1}\circ g|_{g^{-1}(f(W))}$ maps a neighborhood of $0\in H^k$ to a neighborhood of $0\in H^k$. Then $\phi_*(e_k)$ is a vector in the (strict) upper half-space, as you can check. This tells you that the notions of "outward-pointing" will coincide for both parametrizations. (Indeed, this shows that $\phi_*$ maps $H^k$ to $H^k$.) And you use what we've just said about $\phi$ to conclude, with the chain rule, that since $g=f\circ\phi$ (with domain appropriately restricted), we have $f_*(H^k)\subset g_*(H^k)$. Now just use the symmetric argument to get the reverse inclusion.
Remember that the consistency check for orientations is only done on a (connected) open chart. Since the orientation on $M$ is consistent, and since $n(x)$ is the image of a vector with negative $k^{\text{th}}$ coordinate for all $x$ in that chart, the result is immediate.