In the integral $\iiint y^3x^2\,dx^3$ what does the $dx^3$ mean? I interpret it as being shorthand for $dx\,dx\,dx$ because you need three differentials for a triple integral and since the $dx$ is a single variable. But I'm not very familiar with triple integrals and that's why I'm asking here.

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    $\begingroup$ @user326159 I'd say that the OP's interpretation is correct, usually $dx^3 = dx\,dx\,dx$. $\endgroup$ Feb 25, 2021 at 12:14
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    $\begingroup$ @user326159 if we were to be using $dxdydz$ why would that be shortened to $dx^3$? $\endgroup$
    – Marin
    Feb 25, 2021 at 12:17
  • $\begingroup$ $dx^3$ can also mean $3x^2dx$ in the case of an integral, but it doesn't seem to be right in the context of a triple integral. $\endgroup$
    – MasB
    Feb 25, 2021 at 13:07
  • $\begingroup$ It might help to see more of the context in which this came up. Will the integrals be made definite? Or is it possible that we're looking at the third antiderivative of a constant $y^3$ times $x^2$? $\endgroup$
    – David K
    Feb 25, 2021 at 13:08
  • $\begingroup$ Yep, that would be noted $dV$ for $dxdydz$ as a volume infinitesimal, or maybe it was meant for $\mathbf x=(x,y,z)$ with a bold face $x$ on LHS, may happens in books composition. Nonetheless David K's proposal that it might be a triple anti-derivative rather than a triple integral is also to be considered. $\endgroup$
    – zwim
    Feb 25, 2021 at 13:22


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