In applied mathematics, when can we assume that we are allowed to do this:
$$ \frac{\partial}{\partial x}\int_{x}\int_{y}\cdots\int f(x, y,\cdots)\ dx\ dy\ d(\cdots)= \int_{x}\int_{y}\cdots\int \frac{\partial}{\partial x}\left[f(x, y,\cdots)\right]\ dx\ dy\ d(\cdots),$$
where $f(x,y,\cdots)$ is a general continuous and differentiable function over the domain of the variables $(x, y, \cdots)$?
If we view differentiation and integral here as operators, then a more general question would be about the conditions (or properties) that would make two general operators $T_{1}[\cdot]$ and $T_{2}[\cdot]$ commute?
I am interested here in applying such rules to practical calculations (e.g. in physical sciences), so any relevant practical notes or observations about such conditions would be nice.
Excellent question. Rather this is THE question in a lot of higher mathematics which investigate necessary or sufficient conditions for switching operators such as in
$$\lim_{n \to \infty} \int f_n(x) dx = \int \lim_{n \to \infty} f_n(x) dx = \int f(x) dx$$
Based on your other questions, it appears you're taking first or second year Calculus, where you can always switch derivative and integral. There is hopefully some footnote or something in your book saying that all functions in the text satisfy the conditions for switching.
As for applications, I recall
this may have been briefly mentioned in a finance class involving spreads or options.
this occurs in solving partial differential equations such as solving the heat equation using Fourier series.
For example, a solution to the heat equation $u_t = u_{xx}$
could be
$$u=\sum_{n=1}^{\infty} \frac{2}{L} \int_0^L f(x) \sin(\frac{n \pi x}{L}) dx \sin(\frac{n\pi x}{L}) e^{-\frac{n^2 \pi^2 t}{L^2}}$$
Now to begin to differentiate this with respect to either $x$ or $t$, we have to justify $$\frac{\partial}{\partial x} \sum = \sum \frac{\partial}{\partial x}$$
Note that you don't technically switch any $\partial$ and $\int$ because the $\int$ there is actually just a function of $n$ and $L$ and not $x$ or $t$.
As for real world applications: "World Annihilation"