Why does the Jacobian have constant sign for connected sets?
I've seen in two separate proofs now (having to do with manifold orientation) that the Jacobian has constant sign for a connected set, but I can't seem to figure out why that's the case.
One of the proofs I am trying to follow goes like this (From Munkres Analysis on Manifolds):
"We must show $M$ may be covered by such coordinate patches. Given $p \in M$, let $\alpha: U \rightarrow V$ be a coordinate patch about p. Now $U$ is open in either $\mathbb{R}^n$ or $\mathbb{H}^n$; by shrinking $U$ if necessary, we can assume that $U$ is either an open $\epsilon$-ball or the intersection with $\mathbb{H}^n$ of an open $\epsilon$-ball. In either case, $U$ is connected, so $\det{D\alpha}$ is either positive or negative on all of $U$..."
Here $M$ is an n-dimensional manifold in $\mathbb{R}^n$ and $\alpha: U \rightarrow V$ is a diffeomorphism.
There is more after this, but the only part I do not understand is the part bolded.