What kind of integral are we dealing with when we compute the electric field induced by a continuous distribution of charges? [closed]

Question

I am a graduate student in mathematics who has just begun his journey through Griffiths' Introduction to Electrodynamics (it is my first exposure to the subject). I am familiar with differential geometry (manifolds, (pseudo)forms, etc.) and I have some (vague; learning more in the meanwhile) acquaintance with Schwartz distributions. I would like to make sense of what the author is trying to achieve in the languages I already know, but I find even the first pages of the book not easy to follow.

I read about Coulomb's force $$ \vec F = q\,\frac{q^\prime}{\epsilon_0\, 4\pi\, \lVert x - x^\prime\rVert^2}\frac{x - x^\prime}{\lVert x - x^\prime\rVert} $$ exerted by a source (point) charge $ q^\prime $ located at some point $ x^\prime $ on a probe (point) charge $ q $ located at some other point $ x $ and about it's generalization $$ \vec F = \sum_{i = 1}^N q\,\frac{q_i^\prime}{\epsilon_0\, 4\pi\, \lVert x - x_i^\prime\rVert^2}\frac{x - x_i^\prime}{\lVert x - x_i^\prime\rVert} $$ to a bunch of source (point) charges $ q_1^\prime,\dots,q_N^\prime $ located respectively at $ x_1^\prime,\dots,x_N^\prime $. Afterwards I got introduced to the quantity $$ \vec E(x) = \sum_{i = 1}^N \frac{q_i^\prime}{\epsilon_0\, 4\pi\, \lVert x - x_i^\prime\rVert^2}\frac{x - x_i^\prime}{\lVert x - x_i^\prime\rVert}\label{discrete}\tag{$ * $} $$ called the electric field at the point $ x $ generated by the $ q_i^\prime $s. Everything's fine up to now. Please notice that I'm not following prof. Griffiths notations.

My problems arise when "charges distributed continuously over some region" pops out, and the expression for the electric field has to be derived in this case. The author lightheartedly claims that the sum $ \eqref{discrete} $ becomes the (formal?) integral $$ \vec E(x) = \int\frac{\mathrm dq}{\epsilon_0\, 4\pi\, \lVert \vec r\rVert^2}\frac{\vec r}{\lVert \vec r\rVert}\label{continuous}\tag{$**$} $$ where $ \vec r $ should be something like the displacement vector from the infinitesimal charge $ \mathrm dq $ (?) to $ x $.

Following the book, we learn that the $ \mathrm dq $ above can be "instantiated" in the following three cases:

1. if the charge is spread out along a line with charge-per-unit-length $ \lambda $;
2. if the charge is smeared out over a surface with charge-per-unit-area $ \sigma $;
3. if the charge fills a volume with charge-per-unit-volume $ \rho $.

Question 1. What mathematical object is best suited to describe the empirical idea behind $ \lambda $, $ \sigma $ and $ \rho $? I would say that $ \rho $ is a density defined on $ \mathbb R^3 $, and I would say with some confidence that $ \lambda $ and $ \sigma $ are also densities respectively on a $ 1 $-dimensional and a $ 2 $-dimensional submanifold of $ \mathbb R^3 $. Could all of them be thought as some sort of distributions? (I don't know the precise relation between the differential geometer's densities and the analyst's distributions, but I would like to learn more!)

Forgetting for a moment about the mathematical machinery I brought up above, and following again the book, we put

1. $ \mathrm dq = \lambda\,\mathrm dl^\prime $;
2. $ \mathrm dq = \sigma\,\mathrm da^\prime $;
3. $ \mathrm dq = \rho\,\mathrm d\tau^\prime $;

respectively, where $ \mathrm dl^\prime $ is the "element of length" (of our $ 1 $-dimensional submanifold?), $ \mathrm da^\prime $ is the "element of area" (of our $ 2 $-dimensional submanifold?) and $ \mathrm d\tau^\prime $ is the "element of volume" (of $ \mathbb R^3 $?).

If we stuck with $ \lambda $, $ \sigma $ and $ \rho $ being densities, we should probably think of the expressions $ \lambda\,\mathrm dl^\prime $ etc. in the following way: $ \lambda $ as a density can be written as $ \lambda = f\,\mathrm dl^\prime $, where $ f $ is a real valued function and $ \mathrm dl^\prime $ is the Riemannian density induced on the submanifold in question by it's pullback metric. Here we just call $ f $ again $ \lambda $.

Going on, $ \eqref{continuous} $ become $$ \vec E(x) = \underbrace{\int\frac{\lambda\,\mathrm dl^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange line integral?}}\qquad \vec E(x) = \underbrace{\int\frac{\sigma\,\mathrm da^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange surface integral?}}\qquad \vec E(x) = \underbrace{\int\frac{\rho\,\mathrm d\tau^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange volume integral?}}\label{int}\tag{$***$} $$ respectively in the first, in the second and in the third scenario.

Question 2. What mathematical operation do the integrals in $ \eqref{int} $ denote? The equations above really don't make sense at all to me. I'm mostly bothered by two things:

1. if we were really (and sloppily) dealing with tensor densities, what we should get by integrating them are scalars. The integrals appearing above results in vector quantities and this instills me doubts about the nature of $ \lambda $, $ \sigma $ and $ \rho $.
2. assume we are integrating densities: it is not clear if the integrals in $ \eqref{int} $ are meant to denote the the "$ \int D $" (integral of a density $ D $) thing or the "$ \int f(x)\,\mathrm dx $" (Lebesgue integral of the function expressing the density $ D $ as $ D = f(x)\mu $ where $ f $ is a function and $ \mu $ is the Riemannian density induced by the metric) thing. I'll go for the second but I'm still confused.

---

Let's assume for the moment that I made sense of such an expression $ \vec E(x) = \int\text{stuff} $. After all of this there's another question I would like to ask.

Question 3. I have read that the electric and the magnetic fields are best rendered mathematically as respectively a (pseudo-)$ 1 $-form and a (pseudo-)$ 2 $-form. This has something to do with how the electric or the magnetic field components change when we make a change to our reference frame. Now, since we're basically working in $ \mathbb R^3 $ with the standard flat metric, it's obvious how to obtain a $ 1 $-form (or a $ 2 $-form, eventually) from a vector field $ \vec E $. But... this doesn't feel elegant enough to me. What if I wanted to deal with $ 1 $-forms directly? What would these computation look like in this case?

In other (maybe a little silly) words: what is the mathematical object that "integrated" (whatever that means) gives a $ 1 $-form? What is the mathematical object that "integrated" (whatever that means) gives the $ 1 $-form corresponding to $ \vec E $?

---

I would like to emphasize that this question is not about the physics, but about how to talk about the physics in modern mathematical terms. Fore reference, I'm roughly familiar with the material presented in Naber's two volumes book Topology, Geometry and Gauge Fields.

Answer 1

unfortunately I didn't touch any modern-analysis or diff-geometry for a decade, so my answer will be way more "hand-wavy" than the great answer written before me (I guess that's kind of all-right in a physics forum).

My little contribution to the conversation will to add that your already encountered these kinds of sums in probability theory: when dealing with triplets of continuous random variables $(X,Y,Z)$, when you want to calculate moments/averages/other functions you need to use $\iiint pdf(x,y,z)dxdydz$.

But, if there is a deterministic function between two variables (say $y=x^2$) then of course the probability density function is ill-defined in that manner because it is "infinite" when $y=x^2$ and it is $0$ when $y\neq x^2$. This is absurd.

The correct way to calculate functions of the distribution will to look at the manifold parametrized by $(x,z)$ and to calculate a new pdf, one that is defined per unit area and not unit volume.

$\rho , \sigma , \lambda$ are just "slang" for physicists are they dealing with a density which is will-defined on a 3d/2d/1d-manifold in $\mathbb{R}^3$. So to answer your first (and second?) question, take the precise definitions you know from probability theory and apply them here, it's the same concept.

About the third, you can think about each equation you wrote as three different equations, one for each component of the field. For example:

$$E_x (x',y',z') = \iiint \frac{1}{4\pi\varepsilon_0 }\frac{\rho(x,y,z)}{(x-x')^2+(y-y')^2+(z-z')^2} \frac{x'-x}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydz$$

$$E_y (x',y',z') = \iiint \frac{1}{4\pi\varepsilon_0 }\frac{\rho(x,y,z)}{(x-x')^2+(y-y')^2+(z-z')^2} \frac{y'-y}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydz$$

$$E_z (x',y',z') = \iiint \frac{1}{4\pi\varepsilon_0 }\frac{\rho(x,y,z)}{(x-x')^2+(y-y')^2+(z-z')^2} \frac{z'-z}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydz$$

If you formally write $\vec{r} = (x'-x)\hat{i} + (y'-y)\hat{j} + (z'-z)\hat{k}$ you can write all three scalar equations in one vector equation:

$$E_x (x',y',z')\hat{i} +E_y (x',y',z')\hat{j}+E_z (x',y',z')\hat{k} = \iiint \frac{1}{4\pi\varepsilon_0 }\frac{\rho(x,y,z)}{(x-x')^2+(y-y')^2+(z-z')^2} \frac{(x'-x)\hat{i} + (y'-y)\hat{j} + (z'-z)\hat{k}}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydz$$

which is just the equation you wrote: $$ \vec{E} = \iiint \frac{\rho}{||r||^2}\frac{\vec{r}}{||r||} d\tau $$

I hope this helped a bit!

Answer 2

Q3 is actually easiest to answer.

None of those are correct. The only acceptable representation of those fields is as components of the Faraday tensor $F_{ab}$, which is a 2-form on Minkowski spacetime. It is skew-symmetric, so the diagonal is zero, and its time-space parts are the electric field, and space-space parts are the magnetic field.

The Faraday tensor satisfies $\mathrm dF=0$, which motivates us finding a vector potential $A_a$ such that $F=\mathrm dA$, since this immediately leads to $\mathrm dF=\mathrm d\mathrm dA=0$. The price we pay for this convenience, is the inconvenience of having to deal with gauge freedom. The charge and current densities make up a 1-form $j_a$ and they enter Faraday's tensor as $\star\,\mathrm d\star F=j$

---

All of these things are going to just hurt your head, and it is not the fault of anybody trying to be precise. It is just the unfortunateness that physicists have not moved onwards from 19th century mathematics. In particular, because we want to draw vector arrows to visualise everything, so we would not just pretend that the components of the 2-form Faraday tensor are 1-form E and B fields, but that we even take their vectorial duals $\vec E$ and $\vec B$ in order to draw them. All of these things are totally different objects, and the transformation properties of the latter bits are all wrong. This is part of why we have all sorts of nonsense, like "axial vector". It just comes from the fact that physicists think that they can save on lecture time if they do all these imprecise nonsense.

---

Q1 slightly impacts my above answer to Q3. The natural thing to integrate is an n-density, because you can still do integrals on a Möbius strip, even when you cannot properly define a form there. This density has nothing to do with $\rho$ as of yet. The spaces that physics tends to want to play with, have orientation, and thus it is well-defined how to convert from volume forms to their densities and vice versa.

It might seem like we have a charge distribution over some space, so any integration would be over the all available dimensions, so it should be some integral multiple of a volume form. i.e. it seems like it would be natural to define the density, as in physics's sense, as a 3-form. But then you would have to always remember to extract the volume-form as separate from the charge density. So, instead, it is customary to define the physics kind of density as the Hodge dual of this thing. This way, the physics density would be a charge/volume that is defined in a coördinate-independent way.

For line and surface charge densities, you have to have a submanifold, and define the "volume-form" for them. For example, in the line case, you have to define the arc-length density, identify a form for it, and then involve its Hodge dual in order to define the line charge density $\lambda$

---

Q2, you need to know that charge is a Lorentz scalar. That is, if $$\mathrm dq=\lambda\,\mathrm d\ell\qquad\bigvee\qquad\sigma\,\mathrm d^2A\qquad\bigvee\qquad\rho\,\mathrm d^3V$$ then this little thing, after integration, is a scalar as it is. The $\frac1{||\vec r||^2}$ is a magnitude scaling factor, and $\frac{\vec r}{||\vec r||}$ is clearly a unit vector purely giving direction.

That means that the electric field $\vec E$, which, mind you, is already unnatural because it really isn't even a 1-form, let alone the vectorial Hodge dual of it that is now directly being obtained by this integral, is at some observation point $x$ a vectorial weighted average of the charge distribution elsewhere. The different vectorial components contribute as separate scalar-like integrals, if you so want to decompose that integral. That is, if you want to pull in Lebesgue, you would have to treat each Cartesian component separately, and then those would correspond to something that you can apply basic measure theory to. Needless to say, it is vastly superior if you could just upgrade measure theory to work in multiple dimensions.

Answer 3

1. Densities.

There are lots of ideas and terms floating around here; some are related, some are not. For example:

• density in the sense of $\rho$ : this can be viewed a Radon-Nikodym derivative of the charge $Q$ (viewed as a finite-signed-measure on $\Bbb{R}^n$, say $n=3$) relative to the Lebesgue measure $\lambda_n$. This is what’s being alluded to in @Ofek Gillon’s answer. See this answer of mine if you need a slightly gentler explanation.
• density in the sense of $\sigma$: this cannot be interpreted as Radon-Nikodym derivatives of the charge $Q$ relative to Lebesgue measure $\lambda_n$ (because if the signed measure $Q$ is supported on a hypersurface $\Sigma$, then barring trivialities like $Q=0$, it cannot be absolutely continuous relative to $\lambda_n$, so the Radon-Nikodym derivative doesn’t exist). However, if $Q$ is supported on a hypersurface $\Sigma$, then what we mean is that $Q$ is absolutely continuous relative to the surface measure $\lambda_{\Sigma}$ on $\Sigma$ (other common notations include $dV_{\Sigma}$ or $dA_{\Sigma}$ or simply $dA$ or $dS$, and in some contexts, $d\sigma$ as well, but I’m already using $\sigma$ for something else…), and so we can define $\sigma:=\frac{dQ}{d\lambda_{\Sigma}}$.
• density in the sense of linear charge density $\lambda$: same idea as my second bullet point, just consider a 1-dimensional embedded submanifold instead of a hypersurface.
• density as in (a Differential geometer’s) scalar-density on a smooth manifold $M$: this is very closely related to the first bullet point. A scalar density is the natural thing to integrate on a smooth manifold (if $M$ happens to be orientable and oriented, then there is an isomorphism from the space of top-forms to the space of densities, so by ‘transport of structure’ one can define an integral of top-forms).

Now, on an oriented semi-Riemannian manifold, one can write the symbol $\int_Mf\,dV_g$, and a-priori this has three meanings: the Lebesgue integral of a function $f\in L^1(M; dV_g)$ relative to the positive measure $dV_g$. Or it could mean the integral of the scalar density $f\,dV_g$ (where $dV_g$ is viewed as a scalar density), or it could mean the integral of the top-form $f\,dV_g$ (with $dV_g$ now being viewed as the volume-form). Of course all of these numbers are equal, the proof being a simple unwinding of the definitions (the only difficulty being recalling precisely all the definitions); see this answer of mine for slightly more remarks.

For all of these, we can easily generalize them by fixing a (finite-dimensional, real, normed) vector space $W$ and letting all these guys take values in $W$ instead of good old $\Bbb{R}$. All these integrals then of course become $W$-valued rather than $\Bbb{R}$-valued. (The theory of vector-valued Lebesgue integrals is pretty standard; in the finite-dimensional case you can use a basis to immediately reduce it to the $\Bbb{R}$-valued case. More generally, see this answer of mine).

With this, you should be well equipped to precisely interpret (in a measure-theoretic or differential-geometric sense) all the integrals which appear in Griffiths (even the discrete sum is an integral with respect to the charge measure $dQ$, but in this case it just consists of a finite sum of weighted Dirac measures, $dQ=\sum_{i=1}^nq_i\delta_{x_i}$).

I should mention the obvious point that if you’re considering a system of charges comprising of ‘various types’, i.e some combination of point, linear, surface, volume charges, then in order to treat them all uniformly, you should just stick to $Q$ as a signed measure; it won’t have a global Radon-Nikodym derivative (relative to either of $\delta$ or $\lambda$ or $\lambda_{\Sigma}$ or $\lambda_{\Bbb{R}^n}$). Having said that, of course, if the support of $Q$ consists of a disjoint sets each of which is of a ‘fixed type’, then you can restrict your measure to each of these types and consider the appropriate Radon-Nikodym derivatives in each region separately.

---

2. Distributions.

• distributions in the sense of Laurent Schwartz: fix a finite signed measure $Q$ on $\Bbb{R}^n$. By integration against test functions, this defines a distribution, still denoted $Q$ by slight abuse of notation, on $\Bbb{R}^n$. So, in this sense, you can view the charge measure as a distribution.
• distribution in the bundle-theoretic sense: not at all related to what we’re discussing.
• $k$-currents in the sense of de Rham/Schwartz: these are generalizations of the analyst’s distributions on open subsets of $\Bbb{R}^n$ to oriented manifolds. The idea is precisely the same though: rather than ‘testing’ against smooth compactly supported functions, a $k$-current is obtained by ‘testing’ against smooth compactly supported $k$-forms (i.e a $k$-current is an element of the dual $(\mathcal{D}_k(M))’$ of the space $\mathcal{D}_k(M)$ of smooth compactly supported $k$-forms on $M$… this space is equipped with a suitable topology making it into a Frechet space). The language of ‘current’ is of course motivated by electrical currents flowing in a thin wire.

We can once again fix a (finite-dimensional, real, normed) vector space $W$, and consider $W$-valued distributions, and $W$-valued $k$-currents on a smooth manifold $M$, etc. Then, when you apply them on your test-functions/test-forms, you get outputs in $W$ rather than $\Bbb{R}$.

---

Some Remarks on the More Conceptual Aspects of the Physics.

Hopefully the two parts above answer some of your questions on the ‘type’ of objects various things can be interpreted as and how they’re related. Now, coming to your question 3, note that as mentioned in @naturallyInconsistent’s answer, the thing we ‘really’ want is not the $E$-field or $B$-field, but rather the full electromagnetic Faraday 2-form $F$ (we only get the $E,B$ fields after fixing a foliation of spacetime into time+space; see Baez, Munian- Gauge Fields, Knots and Gravity, Part 1 for more details).

What the equations $(*),(**),(***)$ you’ve written down are trying to do is solve Poisson’s equation by finding a Green’s function (namely the Coulomb/Newtonian kernel, which is $\pm\frac{1}{4\pi r}$ in 3-dimensions) and convolving against it (we can even dare attempt this only because the PDE is linear). Later on in chapter 5 of Griffiths does similar stuff in the context of magnetostatics; mathematically, nothing different really happens.

The question now becomes how can we generalize this to more dimensions and more general manifolds? Well, recall that in a nice system of units, Maxwell’s equations read \begin{align} \begin{cases} dF&=0\\ \delta F&= J \end{cases} \end{align} where $\delta=d^*$ is the formal adjoint of the exterior derivative (the codifferential), $F$ is the Faraday 2-form (the electromagnetic field strength), and $J$ is the current $(2-1)=1$-form (a-priori not to be confused with the de Rham/Schwartz current introduced above).

One might wonder whether it’s possible to express $F$ directly as an integral involving $J$ over some region and the values of $F,\star F$ on the boundary. For that, let us record the following analogue of the classical Green’s identities (the second and third ones).

Lemma. (Green’s identity)

Let $(M,g)$ be an oriented semi-Riemannian manifold. Introduce the following notation:

• Let $d$ be the exterior derivative, $\delta=d^*$ the codifferential and $\Delta:=d\delta+\delta d$ the Hodge-Laplace operator (it sends $k$-forms to $k$-forms).
• for any open subset $\Omega\subset M$ and any $k$-forms $\alpha,\beta$, let $\langle\alpha,\beta\rangle_{\Omega}:=\int_{\Omega}\alpha\wedge\star\beta$ (assuming the integrals converge… which will be the case if $\text{supp}(\alpha)\cap\text{supp}(\beta)\cap\overline{\Omega}$ is compact)
• for any codimension-1 submanifold $\Sigma$ (e.g $\partial\Omega$ if $\Omega$ is smooth enough) we define the ‘boundary terms’ $B_{\Sigma}(\alpha,\beta):=\int_{\Sigma}(\delta\alpha\wedge\star\beta-\beta\wedge\star d\alpha)$.

Then, for any smooth $k$-forms $\alpha,\beta$ and open set $\Omega$ with nice enough boundary (and such that the converge nicely so that we can apply Stokes’ theorem), we have that \begin{align} \langle\Delta\alpha,\beta\rangle_{\Omega}-\langle\alpha,\Delta\beta\rangle_{\Omega}&= B_{\partial\Omega}(\alpha,\beta)-B_{\partial\Omega}(\beta,\alpha). \end{align}

In ‘particular’, let $\alpha=F$ be any closed $k$-form (for electrodynamics we will want to set $k=2$) and that for some fixed point $p\in \Omega\subset M$, we have that $\beta=G_p$ is a $\bigwedge^k(T_p^*M)$-valued $k$-current on $M$ which is smooth away from $p$ with $G_p$ and its first derivatives integrable near $p$, and which satisfies $\Delta G_p=\delta_p$, i.e it’s a Dirac $k$-current concentrated at the point $p$, meaning that when we test against smooth $k$-forms $\alpha$, we get the value of $\alpha$ at the point $p$, i.e $\alpha_p\in \bigwedge^k(T_p^*M)$. Then, applying the above (and strictly speaking, we need to do some limiting arguments) we have \begin{align} \langle d\delta F +\underbrace{\delta dF}_{=0},G_p\rangle_{\Omega}-F_p&=B_{\partial\Omega}(F,G_p)-B_{\partial\Omega}(G_p,F)\\ &=\int_{\partial \Omega}(\delta F\wedge \star G_p-0)- B_{\partial\Omega}(G_p,F). \end{align} By applying Stokes’ theorem on the LHS, we get \begin{align} \langle\delta F,\delta G_p\rangle_{\Omega}+\int_{\partial\Omega}\left(\delta F\wedge \star G_p\right)- F_p&= \int_{\partial \Omega}\delta F\wedge\star G_p- B_{\partial\Omega}(G_p,F). \end{align} Cancelling the like terms and rearranging, and denoting $J:=\delta F$ yields \begin{align} F_p&=\langle\delta F,\delta G_p\rangle_{\Omega}+B_{\partial\Omega}(G_p,F)\\ &=\int_{\Omega}J\wedge \star \delta G_p+\int_{\partial\Omega}(\delta G_p\wedge \star F-F\wedge \star dG_p).\tag{$*$} \end{align} So, what equation $(*)$ is saying in words is that if we have a $k$-form $F$ which satisfies $dF=0,\delta F=J$ (for some given $(k-1)$-form $J$ on $M$), and assuming we’re able to find such a “Green’s $k$-form” $G_p$ for each point $p$, then we can recover the value of the $k$-form $F$ at the point $p$ by the formula $(*)$, which you should observe involves only the values of $J$ ‘inside’ the region $\Omega$, and the values of $F$ and $\star F$ on the boundary $\partial\Omega$. See Thirring’s Classical Mathematical Physics for such a presentation.

Something I glossed over here is why such a Green’s $k$-form (a generalization of Green’s function for Poisson’s equation above) $G_p$ exists. Well, this is not trivial. For wave equations (which is ultimately what Maxwell’s equations in $(n+1)$ Lorentzian manifolds boils down to), see the treatment in Friedlander - The Wave Equation on a Curved Spacetime, and also see Bar, Ginoux, Pfaffle - Wave Equations on Lorentzian Manifolds and Quantization for all the details I skipped.

Finally, I must say that none of this will be intelligible if you don’t have some basic intuition from Griffiths, or some other more basic experience with the flat scalar wave equation. A lot of the general theory takes ideas and inspiration from those two model problems and goes crazy with generalizations.