My prelims recommendations
As far as general physics knowledge goes, my list of textbooks is pretty standard: Thornton and Marion and Goldstein for Mechanics, Griffiths and Zangwill for E&M, Townsend, Griffiths, and Shankar for Quantum, and Tong’s notes for Statmech.
Ross Dempsey and Andy Leifer have put together Undergraduate Physics in a Hurry, a really well-typeset prep guide which is “one-third exposition, one-third worked prelim problems, and one-third bad jokes” – the worked examples here in particular are incredibly valuable.
You should also be aware of this dropbox contributed and maintained by some of the older grad students, as well as the solutions website.
There are some other heirloom Princeton resources which should be emailed out to you at some point, but this is my attempt at centralizing some of them.
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}