Differential Vector Element I am a little confused about differential element notations. Say we have a function $f$ defined on a 3D space. Would the following notations be identical?
$$\int f(\vec{x}) \, d\vec{x}$$
$$\int f(x_{1},x_{2},x_{3}) \, d^3x$$
Intuitively the equivalence makes sense: if we sum over all combinations of $x_{1},x_{2},x_{3}$, we sum over all possible vectors. But notation wise how can we convert $d\vec{x}$ to $d^3x$. In certain cases would it be better to use one notation over the other?
 A: I have never seen $d^3x$ in this context and TBH I'm not sure it's accurate.  Technically it would be $(dx)^3$ but even that isn't correct because your variables in the second form are $x_1, x_2, x_3$.
What I've seen and what I use is $$ \iiint f(x_1, x_2, x_3) \, dx_1 \, dx_2 \, dx_3.$$
It's also possible, although I think uncommon and perhaps informal, to say $$\displaystyle \int f(x_1,x_2,x_3) \, dx_1\, dx_2\, dx_3.$$  I think I've only seen this in the context of measure theory.

But notation wise how can we convert $d\vec{x}$ to $d^3x$.

If $\vec x = \langle x_1, x_2, x_3 \rangle$, then $d\vec x$ can be replaced with $dx_1 \, dx_2 \, dx_3$ in the integral.  For definite integrals you'll need to be careful about the order in which those three are written.

In certain cases would it be better to use one notation over the other?

It's really just personal preference as to which one you use, but I think the form with $x_1, x_2, x_3$ is generally better if you want to emphasize or highlight either the three different variables in use or the order in which the integrations are to be done.
