Stupid math tricks

(a.k.a. mathematical methods you were never taught)

Find below collected wisdom from 4+2 years of attempting problem sets.


Multivariate delta functions

\[\int \delta({g(\vec{x})}) d\vec{x} = \Sigma_{\text{zeros of} g(\vec{x})} \frac{1}{|\nabla g(\vec{x})|}\] Plug this into your integrand and simplify before evaluating.

Completing the square to integrate

Integrals that Mathematica won’t do

When is it a hyperbolic trig function?

Elliptic integrals

Example: computing an orbital period

Abel transform

Example: projecting onto the sky in the distant observer approximation

Fourier transform conventions

Newton’s shell theorems

Useful variable substitutions

Similarity variables

Perturbation expansions

Mixed-order expansions

Sturm-Liouville Equations

Example: stellar pulsations

Coulomb Logarithms

Example: Chandrasekhar dynamical friction formula