# Show that the boundary orientation agrees with the standard orientation of $\mathbb R^{k-1}$ if and only if k is even.

$H^{k}$ is oriented by the standard orientation of $\mathbb R^{k}$. Thus $\partial$$H^{k} acquires a boundary orientation. But \partial$$H^{k}$ may be identified with $\mathbb R^{k-1}$. Show that the boundary orientation agrees with the standard orientation of $\mathbb R^{k-1}$ if and only if k is even. Symbolically, we may express this as $\partial$$H^{k} = (-1)^{k}$$\mathbb R^{k-1}$.

This is Question 3.2 #6 from Guillemin and Pollack Differential Topology. If $\partial$$H^{k}$ may be identified with $\mathbb R^{k-1}$, why don't they have the same orientation? Aren't they both just represented by {$e_1$,...$e_{k-1}$}?