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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.

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    The assertion that *these things are mysteriously not taught anywhere* is trivially wrong. It might be the case that some physics curricula skip over them too quickly but, for example, the two first identities you mention are a standard part of most (maths) differential calculus courses I am familiar with. The third identity is odd, and begs for some clarification (what is $\dot f$?).2012-03-21
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    @DidierPiau fixed. http://en.wikipedia.org/wiki/Newton%27s_notation . The dot signifies a time-derivative.2012-03-21
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    I think that at least part of the reason why these rules are often found confusing is the persistent abuse of notation that goes on, for example when we write $x=x(t)$ (using the same symbol for a variable and a function). Clearly it makes sense to differentiate with respect to the *variable* $x$, but (in this context) it doesn't make sense to differentiate with respect to the *function* $x$. You might find it easier if you write down these formulas using $x=X(t)$ or something similar, to make it clear when you are dealing with a function and when you are dealing with a variable.2012-03-21

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