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When writing a multiple integral, there is sometimes used a shorthand for writing the differential in the integral.

For example in $\mathbb{R}^3$ instead of writing $\mathrm{d}x\ \mathrm{d}y\ \mathrm{d}z$ we sometimes find $\mathrm{d}^3 \bf{x}$ where $\bf{x} \in \mathbb{R}^3$, or even $\mathrm{d}^3x$. Where does this come from?

I assume this is a completely separate notation from a similar one which might indicate a third order differential, e.g. in the expression $\frac{\mathrm{d}^3y}{\mathrm{d}x^3}$, as it does not make sense to consider $\mathrm{d}^3 \bf{x}$ as something like "the $\mathrm{d}$ operator applied three times to x", even though it is a sort of "third order" object used in the multiple integral (as it is $\mathrm{d}x\ \mathrm{d}y\ \mathrm{d}z$).

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    $\begingroup$ One also sometimes sees (especially in physics) the volume element written as $d\mathbf{x}$. $\endgroup$ – Semiclassical Oct 29 '14 at 12:17
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The notations $dV$, $d^3\mathbf{x}$, and $d^3x$ are usually all equivalent; they all denote the standard volume element on $\mathbb{R}^3$. In rectangular coordinates this is $dx\,dy\,dz$, but in cylindrical coordinates this is $r\,dr\,d\theta\,dz$ and in spherical coordinates this is $\rho^2\sin\phi\,d\rho\,d\phi\,dz$.

For example if $S$ is the unit sphere then $$\iiint_S dV = \iiint_S d^3\mathbf{x}=\iiint_S d^3x$$ is an integral whose value is the volume of $S$. If we want to actually compute the volume, we choose some coordinates and use iterated integrals: for example $$\int_0^{2\pi}\int_0^\pi\int_0^1 \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$$ in spherical coordinates or $$\int_0^{2\pi}\int_0^1\int_{-\sqrt{1-r^2}}^{+\sqrt{1-r^2}} r\,dz\,dr\,d\theta$$ in cylindrical coordinates or $$\int_{-1}^{+1}\int_{-\sqrt{1-x^2}}^{+\sqrt{1-x^2}}\int_{-\sqrt{1-x^2-y^2}}^{+\sqrt{1-x^2-y^2}}dx\,dy\,dx$$ in rectangular coordinates.

The $3$ in $d^3\mathbf{x}$ and $d^3x$ is just there as a reminder of the dimension; if we were integrating in $\mathbb{R}^4$ the volume element would be $d^4\mathbf{x}$ or $d^4x$. Whether the $x$ is boldfaced or not is just a matter of convention.

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    $\begingroup$ Thanks for your answer, I know that the $d^3 x$ stands for the volume in all these various coordinate systems, I just chose dimension 3 as an example really. From your answer I can see that you are saying $d^n x$ has nothing to do with higher-order differentials. But I would be interested to know where this comes from and who originally suggested this notation as it is easy to confuse with a higher-order differential. $\endgroup$ – user50229 Oct 29 '14 at 14:17

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