Thread by @robertghrist

teaching green's theorem, gauss's theorem, & stokes' theorem always prompts the question, "what is this good for?" although fluids & emag are important, it would be nice to have a more modern application... how about to data?
1/13

classical applications of green's to area have a modern update: consider a swarm of drones that detect the boundary of a region (say, a forest or a forest-fire). can they estimate the area in a decentralized manner? yes, and the formula is a path integral over the boundary.
2/13

ok, so what about gauss? ah, well, the same idea works to compute volume based on vertices of a triangulated surface in 3-d, and that is especially nice for medical imaging, where nonconvex bodies are rather common...
3/13

now, all this is classical (cf. planimeters), if not as well-known as it should be. in addition, one can compute centroids, moments, and more based on point-cloud data along the boundary of a domain in 2-d or 3-d. nice!
4/13

ok, but what about stokes' theorem? i'm so glad you asked... here's something new (as far as i am aware...) consider the problem of estimation of surface area on a sphere given lat/long coordinates of boundary points...
5/13

the key is whether the surface area 2-form is exact -- is it the derivative of a 1-form? yes, yes it is! that means stokes' theorem gives a formula estimating area on the sphere based on boundary point data.
6/13

i don't think this formula appears previously anywhere in the literature. in practice, most scientists tend to do an area-preserving map to the plane, then use green's theorem. and that's fine...
7/13

but one of the reasons why this formula was not previously obvious is that everything is clear when using differential forms, but few people use that toolset. if you try to get this result using classical vector-calculus notation... well... it seems awfully confusing to me
8/13

ah, but there are some pitfalls... how does this formula know which part of the sphere you are trying to compute the area of? (hello, jordan curve theorem)
9/13

and, it gets worse -- if you try to use this formula to compute a spherical cap about the north pole, you get a negative answer. oops! oh, right, the 1-form field is singular at the north and south poles...
10/13

rather than being bad news, it's all good, as this provides an excuse to start talking to the curious student about the topology of a sphere, the (hidden) euler characteristic concentrated at the singular poles, and de rham cohomology...
11/13

...not that i would do that in a calculus class, mind you.
but! with stokes' theorem, the calculus student comes close to touching the stars...
12/13

if you want to show your students these applications, you'll find them in chapter 15 of volume 4 of calculus BLUE.
13/13