While trying to understand certain properties of elliptic curves recently, I realized two interesting things that follow from the Stokes' theorem. This is a generalization of the fundamental theorem of calculus that tells us that if D is a region with boundary ∂D then we have that
∫D df = ∫∂D f
The first thing that I stumbled across relates to integrating z* where we're writing z* for the complex conjugate of z because the usual notation of a bar over the z looks really bad in HTML.
Let's suppose that C is a positively-oriented loop in the complex plane that encloses an area of A. We have that
∫C z* dz = ∫C (x – i y) (dx + i dy)
= ∫C x dx + y dy + i x dy - i y dx
But because C is a loop, we have that
∫C x dx = ∫C y dy = 0
so that we're left with
∫C z* dz = i ∫C x dy - y dx
But from Green's theorem, which is a special case of Stokes' theorem (and is something that everyone learns in Calculus III but almost everyone promptly forgets), we know that
∫C x dy - y dx = 2A
so that we find that
∫C z* dz = 2iA
That's something that I hadn't seen before. It's not exactly profound, but it's sort of interesting.
The second thing that I realized follows from Stokes' theorem is integration by parts. If we write f = u v then we have that
∫D d(u v) = ∫∂D u v
∫D u dv + v du = ∫∂D u v
∫D u dv = ∫∂D u v – ∫D v du
When D is just an interval [a,b] in the real numbers, that's just another way of writing the usual integration by parts formula that we see in Calculus I. That's also a point of view that I hadn't seen before and is just as non-profound-yet-interesting as the first observation. It's also the sort of thing that should probably be mentioned in your typical Calculus III class. It's definitely wasn't mentioned in mine.
Unfortunately, thinking about these things didn't help me understand what I was trying to understand about elliptic curves, but I don't think that the 10 or 15 minutes that I spent on this digression was totally wasted. It's always good to see how everything fits together.