The Golden Rule
Other math facts!
vector calc thing i always forget: \begin{equation} \nabla\times(\nabla\times \vb{A}) = \nabla(\nabla\cdot\vb{A}) - \nabla^2\vb{A} \end{equation}
a tricky taylor expansion: \begin{equation} \frac{1}{|a-r|} = \frac{1}{\sqrt{a^2-2a\cdot r+r^2}} = \frac{1}{|a|}\frac{1}{\sqrt{12-2a\cdot r/a^2+r^2/a^2}} \approx \frac{1}{|a|}\qty(1+\frac{r\cdot a}{a^2}) \end{equation}
common integral: \begin{equation} \int_0^{\infty}\dd{x}x^ne^{-x} = n! \end{equation}
Stirling’s approximation: \begin{equation} \log N! \approx N\log N - N \end{equation}
Laplace’s method/ saddle point integration: \begin{equation} \int h(x) e^{Mg(x)} \approx \sqrt{\frac{2\pi}{M|g’‘(x_0)|}}h(x_0)e^{Mg(x_0)} \end{equation} for $M$ large and $x_0$ the location of the maximum of $g$
Spherical coordinates \begin{equation} \nabla^2 f = \frac{1}{r^2}\pdv{r}\qty(r^2\pdv{f}{r}) + \frac{1}{r^2\sin\theta}\pdv{\theta}\qty(\sin\theta\pdv{f}{\theta}) + \frac{1}{r^2\sin^2\theta}\pdv[2]{f}{\phi} \end{equation}
🅱aussian \begin{equation} \int\dd[n]{x} \exp\Big(- \frac{1}{2}\vb{x}\cdot\mathbb{A}\cdot\vb{x} + \vb{v}\cdot\vb{x}\Big) = \sqrt{\frac{(2\pi)^n}{\det\mathbb{A}}}\exp\Big(\frac{1}{2}\vb{v}\cdot\mathbb{A}^{-1}\cdot\vb{v}\Big) \end{equation}