Many times I come across some new formula being used to work with and/or reduce partial differentials. As kleingordon said, these things are mysteriously not taught anywhere(atleast in physics courses). I can't find any list on the internet, either.
I'm talking about formulae like these:
$\frac{\mathrm{d}}{\mathrm{d}\alpha}\int f(x,\alpha) \mathrm{d}x=\int\frac{\partial f(x,\alpha)}{\partial \alpha}\mathrm{d}x$
$\frac{\partial}{\partial x}\frac{\partial f}{\partial y}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}$ (for continuous functions)
I've also seen that you can stuff a derivative inside a PD $ \frac{\rm d}{\rm dt}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial \dot f}{\partial x}$ (Note-$\dot f=\frac{\rm df}{\rm dt}$)
There's also a formula that allows one to split a function into a sum of partial derivatives. I think this is the multivariable chain rule.
I'd like a list of such formulae, or links to these lists. Books are also fine, though I'd prfer internet sources.