Notation of a volume integral

Question

I see a physics textbook that switches between the following three notations:

\begin{equation} \int_V \text{d}V \end{equation} \begin{equation} \int_V \text{d}^3\vec x \end{equation} \begin{equation} \int_V \text{d}^3 x \end{equation}

where all three are meant to be equivalent. The quantity $V$ is the volume of a sphere and $\vec x$ is a position vector.

The latter two seem to be in contradiction. Which is correct? Or are all three fine?

Answer 1

They are not contradictory, it is just a notation to denote a volume element, just keep that in mind, here are some other notations I've seen around

• $\displaystyle{\iiint_V {\rm d}^{3}V}$

• $\displaystyle{\iiint_V {\rm d}x{\rm d}y{\rm d}z}$

• $\displaystyle{\int_V {\rm d}^3{\bf r}}$

As long as you are clear on the meaning of all these symbols: volume integral over the region $V$, there shouldn't be any problem

Answer 2

Since you mentioned that you were reading a physics book, I suppose there are at least three aspects involved.

Firstly, physicists usually state the laws of physics first before worrying about computation. Laws of physics are valid for the space in which phenomena occur. A standard model for this space is Euclidean space $\mathbb{E}^N$. In order to compute, you have to use a chart $C \subset \mathbb{R}^N$ with coordinates. $\mathbb{R}^N$ is a particular Euclidean space in which computations can be performed because points are labelled by numbers.

A typical example is the balance of mass. A physicist might express in a math-like notation, that conservation of mass is assumed by writing $$ \frac{\mathrm{d}}{\mathrm{d} t} \int \limits_{\mathcal{B} }\bar{\rho} \mathrm{d} V \,= 0 . $$ Even what looks like a "time derivative" is at this point only a declaration of interest called the material "time derivative".

For computation, you have to use a chart and then you may write down in coordinates how to compute the mass of the object, e.g., $$ M= \int \int \int_\Omega \rho(x_1, x_2, x_3) \, \mathrm{d} x_1 \mathrm{d} x_2 \mathrm{d} x_3 \, . $$ using a particular chart with coordinates $x_1, x_2$ and $x_3$.

Secondly, as already pointed out by caverac, different notations are possible and notation is eventually also a matter of taste. But, notation matters a lot because it determines how much time is needed to understand a problem. Therefore, regarding $\mathbb{R}^N$, I think $$ \int_\Omega \rho \, \mathrm{d}^N \, x $$ is a good choice because it is short, flexible and not ambiguous:

• you know from $N$, that $\Omega \subset \mathbb{R}^N$ and,
• obviously, you need coordinates $x_i, i=1,..N$.

Thirdly, if, for instance, integration theorems are involved, many people prefer a notation like $$ \int_\Omega \nabla \cdot \vec{u} \, \mathrm{d} V = \int_{\partial \Omega} \, \vec{u} \cdot \vec{n} \, \mathrm{d} S $$ in order to avoid a discussion about the technicalities of integration over nonlinear surfaces in $\mathbb{R}^3$ at a point where it only might distract the reader from what authors consider important at this moment.