At last we can state one of our special cases of Stokes’ theorem. We don’t need to prove it, since we already did in far more generality, but it should help put our feet on some familiar ground.
So, we take an oriented curve in a manifold . The image of is an oriented submanifold of , where the “orientation” means picking one of the two possible tangent directions at each point along the image of the curve. As a high-level view, we can characterize the orientation as the direction that we traverse the curve, from one “starting” endpoint to the other “ending” endpoint.
Given any -form on the image of — in particular, given an defined on — we can define the line integral of over . We already have a way of evaluating line integrals: pull the -form back to the parameter interval of and integrate there as usual. But now we want to use Stokes’ theorem to come up with another way. Let’s write down what it will look like:
where is some -form. That is: a function. This tells us that we can only make this work for “exact” -forms , which can be written in the form for some function .
But if this is the case, then life is beautiful. The (oriented) boundary of is easy: it consists of two -faces corresponding to the two endpoints. The starting point gets a negative orientation while the ending point gets a positive orientation. And so we write
That is, we just evaluate at the two endpoints and subtract the value at the start from the value at the end!
What does this look like when we have a metric and we can rewrite the -form as a vector field ? In this case, is exact if and only if is “conservative”, which just means that for some function . Then we can write
which should look very familiar from multivariable calculus.
We call this the fundamental theorem of line integrals by analogy with the fundamental theorem of calculus. Indeed, if we set this up in the manifold , we get back exactly the second part of the fundamental theorem of calculus back again.