Manifolds with boundary

Question

Let $M$ be the left half space $\{(x^1,\ldots, x^n) \in \mathbb R^n\mid x^1 \le 0\},$ with orientation form $dx^1 \wedge \cdots \wedge dx^n.$ Show that an orientation form for the boundary orientation on $\partial M=\{(0,x^2, \ldots , x^n) \in\mathbb R^n\}$ is $dx^2\wedge \cdots \wedge dx^n.$

Answer 1

It is enough to show that $\omega=dx^2\wedge..\wedge dx^n$ in not degenerated on the boundary. Let $x=(0,x^2,..,x^n)$ an element of the boundary. $e_i, i\geq 1$ the vector whose only non zero coordinate is its $i$ coordinate which is one is tangent to the boundary at $x$, $\omega_x(e_2,..,e_n)=1$.