July 12, 2012 1 Comment
It’s now time to introduce the integral. Again, I’ll assume we already understand real integrals. As you might expect, if , then we will say that is integrable if and are both integrable. For the pedantic, here I suppose I mean Riemann integrable. Anyway, if so, then we can define
This is nice, but not exactly what we want. We want path integrals. If is a path, and is defined on some open set containing the image of , then
This may look mysterious, but I assure you, it isn’t. In fact, there is pretty much no difference between path integrals over , and path integrals over . They’re certainly defined similarly. Anyway, let’s do an example. If is the unit circle for , then
If , we have . Otherwise, the integral is
Also, as one might expect, the “speed” at which you traverse a curve is irrelevant. If you reparameterize the curve, the integral computed is identical.
Theorem 1 If is integrable along a path , and is a reparameterization of , then
Proof: Write . We’ll have , , and , all continuous. Then
Using the “-substitution” , so , we get
There are a few more facts I’m going to take for granted. First, if I integrate along a loop , but I traverse it backwards, then the value of the integral is the opposite. If I traverse two paths, first , then , The integral is the sum of the integrals for each piece separately.