My question is that
Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere
$U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on the sphere with coordinates $x^1,...,x^n$
(a) Find an orientation form on $U$ in terms of $dx^1,...,dx^n$.
(b) Show that the projection map $π : U → \Bbb R^n$,
$π(x^1,...,x^n,x^{n+1}) = (x^1,...,x^n)$, is orientation-preserving if and only if $n$ is even
I think that
For part-a, An orientation form on the closed unit ball is $dx^1 ∧ ··· ∧ dx^{n+1}$ and a smooth outward-pointing vector field on $U$ is $\frac{∂}{∂x^{n+1}}$. By definition, an orientation form on $U$ is the contraction $i_{\frac{∂}{∂x^{n+1}}} (dx^1 ∧···∧dx^{n+1}) = (−1)^ndx^1 ∧···∧dx^n.$
I am asking you that all I wrote for part-a is enough to prove part-a. Is this right? And, I could not do part b. Please can you explain me clearly? Thank you for help. I am so willing to understand this