An Algorithmic Lucidity

Green's theorem says that (subject to some very reasonable conditions that we need not concern ourselves with here) the counterclockwise line integral of the vector field F = [P Q] around the boundary of a region is equal to the double intregral of

over the region itself. It's natural to think of it as a special case of Stokes's theorem in the case of a plane. We can also think of the line integral as the integral of the inner product of the vector field with the unit tangent, leading us to write Green's theorem like this:

But some texts (I have Mardsen and Tromba's Vector Calculus and Stewart's Calculus: Early Transcendentals in my possession; undoubtedly there are others) point out that we can also think of Green's theorem as a special case of the divergence theorem! Suppose we take the integral of the inner product of the vector field with the outward-facing unit normal (instead of the unit tangent)—it turns out that

—which suggests that there's some deep fundamental sense in which Stokes's theorem and the divergence theorem are really just mere surface manifestations of one and the same underlying idea! (I'm told that it's called the generalized Stokes's theorem, but regrettably I don't know the details yet.)

But here's something I thought was pretty. We have these two equations, one of which involves the unit tangent vector, and one of which involves the unit normal vector. (It's actually pointing in the opposite direction of the unit normal that we use to define the Frenet frame, which one might argue is an unfortunate clash of conventions, but whatever.) So my linear-algebraic intuitions say: hey, why not smoosh them together in a linear combination? (Making linear combinations is just what you do when you see orthogonal unit vectors, right?) Then it's like we're interpolating between the two forms of Green's theorem. Specifically, let θ be a constant. Then we have

(component decomposition of some unit vector)

(inner product distributes over vector addition)

(linearity of integrals, bilinearity of inner product)

(vectorized Green's theorem)

(planar curl and divergence)

(linearity of integrals)

Isn't that great!? Nor is it a mere coincidence that the coefficients in the final line resemble the entries of the two-by-two matrix representing a counterclockwise rotation by θ. Ideally, we would then generalize this to an arbitrary number of dimensions and gain true insight into the generalized Stokes's theorem, and thus, a facet of the nature of reality itself.

But I'm not there yet.

"I just got an idea that will make a great blog post ..."

"No, it won't."

"What? I didn't even tell you the idea."

"I'm not denying that you had an idea; rather, I claim that it won't make a great blog post, because you won't write it. You wrote two nontrivial posts last year and then abandoned your blog for six months. Obviously, you're just like all the other millions of bags of human refuse who think of themselves as writers and yet never write. They never change, and you've shown by your behavior that you're one of them, so why should I believe you when you say you're going to write something?"

"Uh ..."

"Aww, are you offended? Is our little soi-disant 'aspiring rationalist' incapable of handling a honest statement of the undeniable truth?"

"No, I'm not offended. I disagree—"

"As expected."

"—but I won't argue with you. Just—stay subscribed to the RSS feed, will you?"

"Oh, but of course. It's not like anything will appear in it."

"..."

"This is the part where you threaten to drown me in a veritable deluge of good writing, and then I continue to mock the pathetic cycle of failure and self-delusion at the center of your existence."

"Let's skip that part. I'll see you later."

"I know what you're trying to do! It won't work!"

"I never said it would work."