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I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s f(x,s)\,dx$$ given some regularity assumptions. I was trying to think of some relevant physical interpretations of this.

One that I camp up with (that I think is rather weak) is: the total force exerted by the walls of a chamber holding a gas is defined by an integral. We might want to ask how that function is changing with respect to some parameter of the gas, so we'd differentiate under the integral.

Anyone have a better one?

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Hi Eric, I suggest to link also to the question you made on math.SE. Also, I think the intuition for the rule is a mathematical matter, but that's just my opinion. – pppqqq Jan 14 '14 at 12:12
@pppqqq While you certainly have a point, it's been my experience that physicists are occosaiinally better equipped to answer some math questions by virtue of a difference of emphases in their educational background, and I don't personally think there's anything wrong with asking those questions on this site. This may be one of those questions. – David H Mar 16 '14 at 6:36
Hi @David H, I agree with you for the whole first part, I just think that this specific question is unlikely to get a particular interesting answer (though I hope to be wrong). – pppqqq Mar 16 '14 at 12:49
I suggest you read Feynman's autobiography, 'Surely You're Joking, Mr. Feynman?' where he mentions computing many integrals using differentiation under the integral sign, which others failed to do so through other, more popular methods, e.g. complex analysis. – JamalS May 22 '14 at 15:57
up vote 1 down vote accepted

For example, consider some water flow in the space, in which the density $\rho(x,t)$ fluctuates in space and in time. You might be interested in how the mass inside some fixed volume $V$ changes over time. The mass is equal to $$ M(t)=\int_V{\rho(x,t)\mathrm{d}x}, $$ therefore the "mass flow rate", using the rule you mentioned, is equal to $$ \frac{\mathrm{d}}{\mathrm{d}t}M(t)=\int_V \partial_t\rho(x,t)\mathrm{d}x. $$
Other examples include energy, probability, momentum or state densities instead of mass density.

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I guess a simple example would be the derivation of the continuity equation.

It will first be useful to define the phase space probability density, denoted by $f=f(q_1,\dots,q_{N},p_1,\dots,p_{N},t) \equiv f(\mathbf{x},t)$, that tells us the probability of finding a system near $(q_1,\dots,q_{N},p_1,\dots,p_{N})$ at a time $t$. Therefore: \begin{equation} \mathrm{d}N = f(\mathbf{x},t) \; \mathrm{d}x \end{equation} relates phase space volume elements to the number of particles $N$ in that volume. In other words, $f$ is the number of volume elements occupying the differential phase space volume $\mathrm{d}x$ and: \begin{equation} N = \int \mathrm{d}x \; f(\mathbf{x},t) \end{equation} gives the number of particles in the ensemble.

Now, remember that the flux through a surface is given by: \begin{equation} j = \int f \mathbf{v} \cdot \mathrm{d} \mathbf{S} \end{equation} where $\mathbf{v}$ denotes the flow velocity of the particles. Subsequently, using the $N$-dimensional divergence theorem, the total flux out of a volume $V$ bounded by $\partial V$ is given by: \begin{equation} \oint_{\partial V} f \mathbf{v} \cdot \mathrm{d} \mathbf{S} = \int_V \mathbf{\nabla}_{\mathbf{x}} \cdot \left( f \mathbf{v} \right) \; \mathrm{d}x \end{equation} where $\mathbf{\nabla}_{\mathbf{x}}$ is the $2N$ dimensional gradient on the phase space: \begin{equation} \mathbf{\nabla}_{\mathbf{x}} = \left(\frac{\partial}{\partial q_1}, \dots , \frac{\partial}{\partial q_{N}},\frac{\partial}{\partial p_1}, \dots , \frac{\partial}{\partial p_{N}} \right) \end{equation} On the other hand, using the rule of differentiation under the integral sign, we can write the rate of decrease of particles in the volume as: \begin{equation} \frac{d N}{d t} = - \frac{d}{d t} \int_V \mathrm{d}x \; f = - \int_V \mathrm{d}x \; \frac{\partial f}{\partial t} \end{equation} Therefore, demanding equality between the fourth equation and the sixth equation gives (because the flux out of the volume must equal the reate of decrease of particles in the volume): \begin{equation} \int_V \mathbf{\nabla}_{\mathbf{x}} \cdot \left( f \mathbf{v} \right) \; \mathrm{d}x = - \int_V \mathrm{d}x \; \frac{\partial f}{\partial t} \end{equation} which is the continuity equation.

We also need to use the "differentiation under the integral sign" if we want to derive probability current in quantum mechanics. However, I have not taken this example, because I do not know your background and if/how much knowledge of QM you have. Also for Ehrenfest theorem in QM we need to use this rule.

Another (more mathematical) example when this is useful is for contour integration (and in particular for Cauchy's integral formula).

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Actually the examples from Quantum mechanics would show how neat (and essential) this trick is. – noir1993 May 22 '14 at 15:59
@AchiralSarkar yeah, I agree with you, but the OP has never responded to the various answers, so I'm not sure if he/she would be interested to read about it. – Hunter May 22 '14 at 19:12
@Hunter Sorry for delayed responses, but I have been coming back to this and reading your wonderful answers. They're very helpful. – Eric Auld Feb 1 '15 at 23:11

Possibly the most ubiquitous use is to calculate the derivative of a functional which is defined as an integral, e.g., of the Lagrangian along a path. Typical application---computing the equations of motion as the euler equations of a problem of the calculus of variations

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