If we orient a manifold by picking an everywhere-nonzero top form , then it induces an orientation on each coordinate patch . Since each one also comes with its own orientation form, we can ask whether they’re compatible or not.
And it’s easy to answer; just calculate
and if the answer is positive then the two are compatible, while if it’s negative then they’re incompatible. But no matter; just swap two of the coordinates and we have a new coordinate map on whose own orientation is compatible with .
This shows that we can find an atlas on all of whose patches have compatible orientations. Given any atlas at all for , either use a coordinate patch as is or swap two of its coordinates depending on whether its native orientation agrees with or not. In fact, if we’re already using a differentiable structure — containing all possible coordinate patches which are (smoothly) compatible with each other — then we just have to throw out all the patches which are (orientably) incompatible with .
The converse, as it happens, is also true: if we can find an atlas for such that for any two patches and the Jacobian of the transition function is everywhere positive on the intersection , then we can find an everywhere-nonzero top form to orient the whole manifold.
Basically, what we want is to patch together enough of the patches’ native orientations to cover the whole manifold. And as usual for this sort of thing, we pick a partition of unity subordinate to our atlas. That is, we have a countable, locally finite collection of functions so that is supported within the patch . Then we define the -form on by
and by outside of . Adding up all the gives us our top form; the sum makes sense because it’s locally finite, and at each point we don’t have to worry about things canceling off because each orientation form is a positive multiple of each other one wherever they’re both nonzero.