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  • Stupid math tricks

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

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

    Integrals

    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