Supplemental Math


I'll also throw out some supplementary material on various mathematical topics that sometimes shows up in the notes. Not every topic is essentially relevant to the physics above. It's safe to say none of it is relevant to the Introductory Physics files. Tensors shows up in the context of Special Relativity and General Relativity. The Complex Analysis stuff shows up intermittently in all the advanced Physics files, as does the Fourier Transform stuff. The Path Integral file shows up in Quantum Mechanics, Quantum Field Theory, Statistical Mechanics, Condensed Matter (a bit). The Stochastic Stuff makes a quick appearance when discussing Brownian Motion in the Stat Mech file. But the Integral Equation stuff is entirely absent - I just think it's cool.
  1. Tensors This file covers the math behind a lot of Special and General Relativity. They are somewhat old fashioned in that they use the terminology of covariant and contravariant vectors, whereas within basically the last 50 years, one speaks of one-forms and such. But this way made more sense to me, especially when I was learning it the first time in the context of Jackson's Electrodynamics book. Note, Einstein summation notation is heavily, and often implicitly, used.
  2. Complex Variables These notes are the very first I ever took ( ❤ ) on the computer. I wrote out the equations in Mathcad (this was like 2001). These are also the only notes not in Word, because I didn't feel like transcribing them into that format - maybe one day. So they're all .pdf's. Anyway, in more advanced physics courses, everywhere you look, you end up needing to evaluate some definite integral that can't be done by finding an anti-derivative: Fourier and Laplace transforms and inverse transforms come saliently to mind (and don't you just hate looking up the inverse transform in tables; I know I do). So it's essential to know how to use the Cauchy Residue theorem, etc. That's what these notes ultimately focus on. There's actually some more of that in the Quantum Mechanics folder.
  3. Fourier Transforms This covers how to convert integrals and products in real space to expressions in Fourier space. It's useful to know especially when one covers Green's Functions and you want to translate GF rules from position space to momentum space.
  4. Path Integrals This one starts with simple 1D Gaussian integrals and proceeds to Gaussian Functional (Path) Integrals. And it covers converting such integrals and their perturbations into (simple) diagrams.
  5. Random Matrices and Stochastic Calculus These are some notes I took on Random Matrices and well, Stochastic Calculus. They're together because I was doing both at the same time and they're somewhat interwined. The Random Matrices notes cover how the measure of a matrix ensemble is defined, how to calculate it, as well how probability distributions of matrices are defined, and how to use a max entropy ansatz to calculate some distributions. The Stochastic Calculus part covers finite processes, Wiener Processes, Ornstein-Uhlenbeck Processes, the Stratonovich and Ito Calculus(es) and how to solve Stochastic Differential Equations.
  6. Integral Equations In my research I had to solve a bunch of integral equations, and so I took a lot of notes on them. It's a fascinating topic.