Math

As promised in part I, we can do a lot of the same things with Schwartz distributions as with classical functions. To see how, we’ll cover derivatives, convolutions, and Fourier transforms of distributions.

Did you always want to know kind of object this weird Dirac delta “function” actually is? Well, it’s a Schwartz distribution. If that doesn’t help much, then keep reading.

There’s a style of teaching mathematics that I really like: stating definitions and theorems as formally as in any textbook, but focusing on informal arguments for why they should be true.

Spherical harmonics are ubiquitous in math and physics, in part because they naturally appear as solutions to several problems; in particular they are the eigenfunctions of the spherical Laplacian and the irreducible representations of SO(3). But why should the solutions to these problems be the same? And why are they called spherical harmonics?

Proving things for object that have a lot of structure can be harder than for object with less structure, simply because the tree of possible proofs is much wider. This is probably why trying to prove a more general case is sometimes a helpful strategy.

“Structure” is a concept that keeps popping up when thinking about mathematics but it’s hard to pin down what it is exactly. I discuss several different perspectives for thinking about it.