Let be a bounded subset whose boundary has measure 0.
Let be a continuous [[202404061628|differential -form]] on .
can be written using standard coordinates as
The integral of over is
Written another way
Diffeomorphism invariance
Let and be open subsets in , and to be an orientation preserving or reversing diffeomorphism.
If is a compactly supported -form on then
Where is when the orientation is preserved, and is when the orientation is reversed.
Note: If you are being very precise, you need to use domains of integration, not just open sets.
However, if is compactly supported on an open set, you can always find a domain of integration over the support such that the closure is contained in the open set.
Proof
We start with the standard coordinates on and on .
Let on to be represented in these coordinates as
Then we have the integral
Consider the change of variables from the integration with .
This gives the infinitesimal change in hyper volume units are related by
Then we have that the integral under the change of variables is
Let be an orientable smooth manifold.
Let be an -form on .
Suppose is compactly supported in the domain of a single smooth chart that is positively or negatively oriented.
The integral of over is
with positive for positively oriented chart, negative otherwise.
Definition on whole manifold
Suppose is an orientable smooth manifold, and is a compactly supported differential form on .
Let be a finite open cover of with domains of smooth charts (that are positively or negatively oriented).
Let be a subordinate partition of unity.
The integral of over is
This makes it so for each then the support of the differential form is compactly supported in so we can use the above definition, summing along all sets in the open cover.
Definition is independent of chart
The integral does not depend on a choice of smooth chart whose domain contains the support of .
Proof
It is enough to prove this for the case where is contained in one chart.
Consider charts and such that .
We have 2 possible cases.
Either both and are both orientation preserving/reversing, or that one preserves while the other reverses.
Case 1, both same sign: We have that is an orientation preserving diffeomorphism, so by using the pullback we have
Independent of partition of unity
The definition of given above does not depend on choice of open cover or partition of unity.
Proof
Let and be a finite open cover and a subordinate partition of unity.
Consider another open cover and a subordinate partition of unity.
The main idea of the proof is to partition into even finer open sets to compute the integral.