Spivak problem on orientations. (A comprehensive introduction to differential geometry) I have a problems doing exercise 16 of chapter 3 (p.98 in my edition) of Spivak's book.
The problem is very simple. Let $M$ be a manifold with boundary, and choose a point $p\in\delta(M)$. Now consider an element $v\in T_p M$ which is not spanned by the vectors on $T_p\delta(M)$, that is, it's last coordinate is non-zero (after good identifications). We say that $v$ is inward pointing if there is a chart $\phi: U\rightarrow \mathbb{H}^n$ ($p\in U$) such that $d_p\phi(v)=(v_1,\dots,v_n)$ where $v_n>0$.
It is asked to show that this is independent on the choice of coordinates (on the chart).
I think that Spivak's idea is to realize first that the subespace of vectors in $T_p\delta M$ is independent on the chart, which can be seen noticing that  if $i:\delta (M)\rightarrow M$ then $d_pi(v_1,\dots,v_{n-1})=(v_1,\dots,v_{n-1},0)\in T_p \mathbb{H}^n$ 
 A: A change of coordinates between charts for the manifold with boundary $M$ has the form $x=(x_1, \cdots,x_n) \mapsto (\phi_1(x), \cdots,\phi_n(x))$, with $x_n, \phi_n(x)\geq0$ 
since $x_n,\phi_n(x)\in \mathbb H_n$.  
The last line of the Jacobian $Jac_a(\phi)$ at a point $a\in \partial \mathbb H_n$ has the form $(0,\cdots , 0,\frac { \partial \phi_n}{\partial x_n}(a))$ :
Indeed,  for $1\leq i\leq n-1$ we have $\frac { \partial \phi_n}{\partial x_i}(a)=0$ by the definition of partial derivatives since $\phi(\partial \mathbb H_n)\subset  \partial \mathbb H_n$ and thus $\:\frac {\phi_n(a+he_i)-\phi_n(a)}{h}=\frac {0-0}{h}=0$.
Similarly  $\frac { \partial \phi_n}{\partial x_n}(a)\geq 0$ because $\:\frac {\phi_n(a+he_n)-\phi_n(a)}{h}=\frac {\phi_n(a+he_n)-0}{h}\gt 0$.
Actually, we must have $\frac { \partial \phi_n}{\partial x_n}(a)\gt 0$ because the Jacobian is invertible.  
The above proves that, given  a tangent vector $v\in T_a(\mathbb H_n)$, its image $w=Jac_a(\phi)(v)$ satisfies $w_n=\frac { \partial \phi_n}{\partial x_n}(a)\cdot v_n$ with $\frac { \partial \phi_n}{\partial x_n}(a)\gt 0$, which shows that outward pointing vectors are preserved by the Jacobian of a change of coordinates  for  $M$ and thus that the notion of outward pointing vector is well-defined at a boundary point of a manifold with boundary .
