Notation for triple integral ($d^3x$) I first saw this notation from physics, e.g. quantum mechanics.
Of course, I didn't know what that meant, and after googling I realised that
$$\iiint f \ dV \equiv \int f \ d^3x \equiv \int \int \int f \ dx \ dy \ dz$$
Question 1. Are those three above really equivalent. I just want to ensure.
Question 2. What is the motivation of the notation $d^3x$?
So, for example, for $$\frac{d^3f}{dx^3}$$ we can recognise that the motivation was to express it in its simpler form: $$\frac{d}{dx}\frac{d}{dx}\frac{df}{dx}=\frac{d}{dx}\frac{d}{dx}\frac{d}{dx}f=(\frac{d}{dx})^3f=\frac{d^3f}{(dx)^3}=\frac{d^3f}{dx^3}$$
From this logic, it feels like, because the denominator of $d^3x/dx^3$ is $dx^3$ which represents $(dx)^3$, that we have to write the triple integral in the form $$\int f \ dx^3$$ but it actually is written as $$\int f \ d^3x.$$
 A: Leibniz notation for derivatives does not mean fractions. There are discussions about this on the most voted questions in this forum (really, sort questions by number of upvotes), so I won't go into this. 
What I will say, though, is that the $n$ in ${\rm d}^nx$ is actually a reference to the dimension of the integral. If you want to be precise with notation, you should actually leave $x$ in regular font  for a scalar and use $\vec{x}$ or $\mathbf{x}$ for a vector, so ${\rm d}^n{\bf x}$ is more precise, in the sense that in $2$ and $3$ dimensions we have that $(x,y) = (x_1,x_2)$ and $(x,y,z) = (x_1,x_2,x_3)$, so that for a function of $n$ variables $$\int_\Omega f({\bf x})\,{\rm d}^n{\bf x}$$is a shorthand for $$\int_{\Omega} f(x_1,\ldots,x_n)\,{\rm d}x_1\,\cdots\,{\rm d}x_n,$$and note the convention (usual in mathematics, as opposed to physics) of using a single $\int$ sign instead of $\iint$ or $\iiint$ even for multivariable integrals. If you see $$\int_\Omega f({\bf x})\,{\rm d}{\bf x}$$instead, it also means the same thing. In particular, in integrals for $n=1$, ${\rm d}^1x$ would be the same as ${\rm d}x$ (but no one writes like this). The symbol ${\rm d}^n{\bf x}$ by itself should not have much meaning, although some people would be justified in denoting the standard volume form in $\Bbb R^n$ by it.
