# Do volume integrals involve bounding surfaces?

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• yucheng
In summary, the author derives the vector potential for a magnetic dipole and free currents using integration by parts and taking into account the surface integral term which arises due to the surface current density. The author also discusses the treatment of surface currents in two different ways, either using generalized functions or directly calculating the surface current using the "surface curl".
yucheng
In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents)

\begin{align}
\vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3} r^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}+\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{M}\left(\vec{r}^{\prime}\right) \times\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d^{3} r^{\prime} (1) \\
&= \frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3} r^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}+\frac{\mu_{0}}{4 \pi} \int_{\tau} \vec{M}\left(\vec{r}^{\prime}\right) \times \vec{\nabla}^{\prime}\left(\frac{1}{\left|\vec{r}-\vec{r}^{\prime}\right|}\right) d^{3} r^{\prime} \\
&= \frac{\mu_{0}}{4 \pi} \int \frac{\vec{J}\left(\vec{r}^{\prime}\right)+\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}\\
&= \frac{\mu_{0}}{4 \pi} \int_{\tau - S^{\prime}} \frac{\vec{J}\left(\vec{r}^{\prime}\right)+\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}+\frac{\mu_{0}}{4 \pi} \oint_{S^{\prime}} \frac{\vec{M}\left(\vec{r}^{\prime}\right) \times d \vec{S}^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}
\end{align}

Using integration by parts, we see that the second term of (2) is
$$\int_{\tau} \vec{M}\left(\vec{r}^{\prime}\right) \times \vec{\nabla}^{\prime}\left(\frac{1}{\left|\vec{r}-\vec{r}^{\prime}\right|}\right) d^{3} r^{\prime} = \int_{\tau} \frac{\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}+\oint_{S^{\prime}} \frac{\vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} \times d \vec{S}^{\prime} \tag{5}$$

However, in (5) since the volume on the LHS is inclusive of the surface as well, isn't the volume on the RHS also inclusive of the surface?

In (3), the surface integral term drops out because we integrate over a larger volume such that the magnetization there is zero.

However, for (3) the author claims that the curl is undefined at the boundaries of the magnetized material, hence it is easier to integrate over the volume of the magnetized integral, (4): the volume integral excluding the surface ##\tau - S^{'}## where it is undefined, and the surface integral taking care of the surface ## S^{'}##. However, why can we exclude the surface from the volume integral (now we have an 'open' volume)?

Does this mean that for

$$\vec{E}(\vec{r})=\frac{1}{4 \pi \varepsilon_{0}} \int_{\tau} \frac{\left[-\vec{\nabla}^{\prime} \cdot \vec{P}\left(\vec{r}^{\prime}\right)\right]\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d^{3} r^{\prime}+\frac{1}{4 \pi \varepsilon_{0}} \oint_{S} \frac{(\vec{P} \cdot \hat{n})\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d S^{\prime}$$
we ought to exclude the bounding surface from the volume integral as well?

Last edited:
It depends a bit on how you define the involved derivatives.

As an example take a homogeneously magnetized sphere, i.e.,
$$\vec{M}(\vec{x})=M \vec{e}_3 \Theta(a-r), \quad r=|\vec{x}|.$$
Then you can calculate the equivalent current density
$$\vec{j}_M=\vec{\nabla} \times \vec{M}.$$
Of course that's defined in the usual sense only for ##r \neq a##, and there ##\vec{j}_M=0##. Physically it's clear that the magnetic field is not 0 everywhere. So the effective current is in fact to be understood as a surface current.

Again there are two possibilities to treat it:

(a) using generalized functions

You can read the rotation of the magnetization as an equation for generalized functions. Then you get
$$\vec{j}_M=\vec{\nabla} \times \vec{M}=\vec{\nabla} \times [M \vec{e}_3 \Theta(a-r)]=-m \vec{e}_3 \times \vec{\nabla} \Theta(a-r)=M \vec{e}_3 \times \frac{\vec{r}}{r} \delta(r-a)=\vec{k} \delta(r-a).$$
Here ##\vec{k}## is the said surface current. Now you can use this distribution-valued current density in Biot-Savart's Law for the vector potential (in Coulomb gauge),
$$\vec{A}(\vec{r})=\mu_0 \int_{\mathbb{R}^3} \mathrm{d}^3 r' \frac{\vec{j}_M(\vec{r}')}{4 \pi |\vec{r}-\vec{r}'|}. \qquad (*)$$

(b) using the "surface curl"

There you directly calculate the surface current density as
$$\text{Curl} \vec{M}=\vec{n} \times (\vec{M}_{>}-\vec{M}_{<})$$
along the surface, i.e., the spherical shell ##r=a##. Here ##\vec{n}## is the surfas-normal vector pointing outwards, i.e., ##\vec{n}=\vec{r}/r##,
$$\vec{k}=\text{Curl} \vec{M} = \vec{n} \times (-M \vec{e}_3) = M \vec{e}_3 \times \frac{\vec{r}}{a}$$
for ##\vec{r}## on the spherical shell. Again with Biot-Savart's Law
$$\vec{A}(\vec{r})=\mu_0 \int_{S_a} \mathrm{d}^2 f' \frac{\vec{k}(\vec{r}')}{4 \pi |\vec{r}-\vec{r}'|}.$$
This you get of course also by integrating out the ##r## integral in (*).

yucheng
Thanks @vanhees71 ! I was actually exploring the use of generalized functions to compute the curl and divergence of piecewise functions, especially those representable as step functions. I got really confused when I tried to identify bound volume/surface charges and currents.

P.S. textbooks should really include this example of using generalized functions

Last edited:
vanhees71 said:
Again there are two possibilities to treat it:
Definitely, I should choose either one, but not both! That is, either use volume currents throughout (by evaluating the curl at the surface using generalized functions), or I separate it into volume and surface currents, evaluating wherever the functions are well behaved (evaluating volume currents at the volume but excluding the surface where it's undefined), right?

vanhees71
Yes, of course you must use the equivalent surface current only once, either be it as a distribution-valued current density or as a surface-current density. Taking both would be double counting.

Delta2 and yucheng
yucheng said:
P.S. textbooks should really include this example of using generalized functions
There is at least one book that takes this approach - I do not know how good it is, though.

https://www.amazon.com/dp/1118034155/?tag=pfamazon01-20

If you have access to a library that has it then it might be worth a look. Otherwise, perhaps the author has written papers on the topic that are available online.

Jason

yucheng and vanhees71
Another great, very modern book on classical E&M, making heavy use of the theory of generalized functions is

K. Lechner, Classical Electrodynamics, Springer International
Publishing AG, Cham (2018),
https://doi.org/10.1007/978-3-319-91809-9

yucheng
jasonRF said:
There is at least one book that takes this approach - I do not know how good it is, though.
It appears very specialized ;)

Of course you can use either approach: The traditional one, where the singularities are treated within the integrals or use the equivalent treatment in terms of generalized functions. Both are completely equivalent.

yucheng said:
Just curious, how did you come to know of this book?
If I recall correctly, I saw it in the library at work. I might have flipped through it, but have never read it.

vanhees71 said:
Another great, very modern book on classical E&M, making heavy use of the theory of generalized functions is
K. Lechner, Classical Electrodynamics, Springer International
It appears that Lechner's book mainly deals with singularities, renormalization, relativitstic EM!

Indeed, and that makes it a great addition to more traditional textbooks!

vanhees71 said:
Indeed, and that makes it a great addition to more traditional textbooks!
Are they not covered in the more traditional books? I mean, it's rare to see one that's mathematically rigorous (in the sense as to avoid singularities etc, literally, being careful)

vanhees71
I think the discussion of the radiation-reaction problem is treated in the most satisfying way in this book.

yucheng

## 1. What is a volume integral?

A volume integral is a mathematical tool used to calculate the total volume of a three-dimensional region or object. It involves integrating a function over a specific volume, typically represented by a triple integral.

## 2. How do volume integrals differ from surface integrals?

Volume integrals involve integrating a function over a three-dimensional region, while surface integrals involve integrating a function over a two-dimensional surface. Volume integrals are used to calculate the total volume of a region, while surface integrals are used to calculate things like surface area or flux.

## 3. Do volume integrals always involve bounding surfaces?

Yes, volume integrals always involve bounding surfaces. This is because the limits of integration in a volume integral are determined by the boundaries of the region being integrated over. These boundaries can be represented by bounding surfaces such as planes, spheres, or cylinders.

## 4. How do you set up a volume integral?

To set up a volume integral, you first need to determine the limits of integration, which are determined by the boundaries of the region. Then, you need to choose the appropriate integrand (function being integrated) and the correct order of integration (depending on the shape of the region). Finally, you can evaluate the integral using standard integration techniques.

## 5. What are some real-world applications of volume integrals?

Volume integrals have many real-world applications in fields such as physics, engineering, and economics. They can be used to calculate things like the mass of an object, the amount of fluid flowing through a pipe, or the total charge in an electric field. They are also used in computer graphics to render three-dimensional images and in finance to calculate the value of financial derivatives.

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