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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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    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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$\def\p{\partial}$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 $t$.

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    Yeah, when I checked out Lagrangian mechanics a while ago, that confused me as well. But this answer only proves the third statement. What I would like is a list of as many such formulae as possible. Or a link to one :)2012-03-21
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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.

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    Used that one as well, but thanks for posting!2012-03-21
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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.

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    This does not answer OP s question2016-10-03