I occasionally fantasize about an “ideal math curriculum” for Physics graduate students, based on my experience, in school and out. Which topics make sense, which don’t, what should count as reasonably expected knowledge, what is actually useful?
There are also some textbooks, several of which having been published after I left school, that I would like to use (actually: like to have used) in the appropriate classes.
It turns out, Caltech has made the Feynman Lectures available online on their website.
Imagine a rod that is initially at temperature $T_1$ and then brought into an environment with a lower temperature $T_0 < T_1$. How quickly does the body cool down? When will it have reached the environment’s temperature? What is the temperature profile throughout the rod, as a function of time?
This is essentially a worked homework set: a complete, step-by-step solution of the diffusion (or heat) equation in one dimension.