I'm new to this exchange so please bear with me regarding notation. I would like to know what the differential operator $d^n$ means as seen in some physics equations. Normally, one would have an integral with a $dx$ or $dy$ (whatever the case might be), but the differentials I am talking about are of the form $(d^n)x$. (please note that I'm aware of those in tensor calculus, of the form $d(x^n)$ - I'm not referring to those.) Please help.

  • 1
    $\begingroup$ Could you give an example of its use? Context will be important. $\endgroup$
    – Simon S
    Jun 6, 2015 at 16:31

1 Answer 1


An example: $d^3x = dxdydz$, In general $d^n x$ is a symbol used for the volume element under integration.

For example $\int f(x_1,x_2,x_3,x_4) d^4 x$ means you have to perform 4 integrals, not just one, over a 4-volume domain.

Quite generally:

$$ d^n x = d x_1 dx_2 \cdots dx_n $$ In this sense $d^n$ is not an operator, but rather a shorthand.

  • $\begingroup$ Sometimes they may even write it with boldface $d^n\mathbf x$ to show $\mathbf x$ is $(x_1,\dots,x_n)$ $\endgroup$
    – GEdgar
    Jun 6, 2015 at 17:21
  • $\begingroup$ I'm not sure if your first example is a bad example for your first, or you misunderstand the meaning of $x$ mean $\mathrm{d}^3 x$; it's only accurate if you are writing $x = (x,y,z)$. Usually, we have some other symbol to denote the tuple of coordinate functions; e.g. $\vec{r} = (x,y,z)$, and then we would write $\mathrm{d}^3 \vec{r} = \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$. $\endgroup$
    – user14972
    Jun 6, 2015 at 17:50
  • $\begingroup$ I think that it is a matter of convention, the OP asked specifically for the meaning of the symbol $d^n x$, the first example was intended to clarify making reference to a commonly used notation $(x,y,z)$. Without further context it is hard to choose a symbol to be used. It is also assumed that he/she will also recognise that $x=x_1,~y=x_2,~z=x_3$ in the last line. $\endgroup$ Jun 6, 2015 at 18:15
  • $\begingroup$ Rogelio Molina explained it well and I understood it. Otherwise, I agree that the context would then dictate notation. $\endgroup$
    – Utonium S
    Jun 6, 2015 at 21:06
  • $\begingroup$ @Hurkyl: I don't think it's uncommon for vector x to equal (x,y,z). If you're teaching novices, of course you would use a different letter to avoid confusion, but otherwise it's not something the average physicist would worry about, IMO. (In QM, IIRC, the convention is to use x for position; using r would be liable to confuse the reader.) $\endgroup$ Jun 6, 2015 at 23:48

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