Formulae for PDEs : Commuting derivatives and/or integrals 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.
 A: $\def\p{\partial}$Here's a proof of your last statement. It uses the chain rule: for functions $x(t)$ and $g(x,t)$ you have
$$\frac{d}{dt} g(x,t) = \frac{\p g}{\p t} + \frac{\p g}{\p x} \frac{dx}{dt} \tag{1}$$
If you take $g=\p f/\p x$, then plugging into (1) gives
$$\frac{d}{dt} \frac{\p f}{\p x} = \frac{\p^2 f}{\p t \p x} + \frac{\p^2 f}{\p x^2} \frac{dx}{dt}$$
On the other hand, if first take $g=f$ and then take the partial derivative with respect to $x$:
$$\frac{\p}{\p x} \frac{df}{dt} = \frac{\p}{\p x} \left( \frac{\p f}{\p t} + \frac{\p f}{\p x} \frac{dx}{dt} \right) = \frac{\p^2 f}{\p x\p t} + \frac{\p^2f}{\p x^2} \frac{dx}{dt}$$
You can compare the right-hand sides of these expressions and see that they are equal (since partial derivatives commute). Therefore
$$\frac{d}{dt} \frac{\p f}{\p x} = \frac{\p}{\p x} \frac{df}{dt}$$
so the partial derivative wrt $x$ commutes with the total derivative wrt $t$.
A: An identity related to the first formula that might come in handy is when you have to differentiate under the integral sign, but the limits of integration are functions of the variable you're differentiating with respect to.  Then:
$F = F(\alpha) = \int_{x_1(\alpha)}^{x_2(\alpha)}f(x,\alpha)dx$
$\frac{dF}{d\alpha} = F'(\alpha) = f(x_2,\alpha)\frac{dx_2}{d \alpha} - f(x_1,\alpha)\frac{dx_1}{d \alpha} +  \int_{x_1(\alpha)}^{x_2(\alpha)}\frac{\partial f(x,\alpha)}{\partial \alpha}dx.$
A: About the last statement:Total derivatives are sum of partial derivatives. So consecutive partial derivative and total derivative means actually sum of several two consecutive partial derivatives. And we know partial derivatives commute. So, total derivative also commute with partial derivative.
