4
votes
3answers
131 views

Integral form of Gauss's law for magnetism from Stokes' theorem?

How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?
4
votes
1answer
96 views

Why is $D$ a $2$-form and $E$ a $1$-form?

Usually in electrostatics we start by introducing the vector field $\mathbf{E}$ representing the electric field due to some charge distribution. Later when we study fields in materials we consider the ...
6
votes
1answer
236 views

Why is the Hodge dual so essential?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric ...
1
vote
1answer
258 views

Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed

I am preparing myself for an exam and I got stuck with the following problem. If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I ...
5
votes
2answers
141 views

Exterior (covariant) derivatives and electromagnetism

I'm porting Maxwell's equations to curved spacetime and am having trouble reconciling the tensor and forms treatments. I think the problem boils down to a misunderstanding on my part concerning the ...
7
votes
3answers
747 views

Maxwell's Equations using Differential Forms

Maxwell's Equations written with usual vector calculus are $$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$ $$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times ...
6
votes
2answers
445 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
7
votes
3answers
318 views

Geometric interpretation of Electromagnetism

For gravity, we have General Relativity, which is a geometric theory for gravitation. Is there a similar analog for Electromagnetism?
1
vote
2answers
96 views

Time evolution of the worldlines of 2 particles

Suppose I have a lab frame that is freely falling in a gravitational field of the Earth -- assume non-homogeneity-- and a uniform constant electric field. There are 2 test particles in the frame -- ...
7
votes
2answers
218 views

Forces as One-Forms and Magnetism

Well, some time ago I've asked here if we should consider representing forces by one-forms. Indeed the idea as, we work with a manifold $M$ and we represent a force by some one-form $F \in ...
6
votes
2answers
336 views

Does Kaluza-Klein Theory Require an Additional Scalar Field?

I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} ...
5
votes
1answer
169 views

Kaluza-Klein Christoffel Symbols

I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part ...
4
votes
2answers
399 views

Electromagnetism for Mathematician

I am trying to find a book on electromagnetism for mathematician (so it has to be rigorous). Preferably a book that extensively uses Stoke's theorem for Maxwell's equations (unlike other books that on ...
6
votes
1answer
845 views

Maxwell's equations in microscopic and macroscopic forms, and quantization

The macroscopic Maxwell's equations can be put in terms of differential forms as $$\mathrm{d}\mathrm{F}=0,\quad\delta \mathrm{D}=j\implies \delta j=0,\quad \mathrm{D}=\mathrm{F}+\mathrm{P}.$$ ...
3
votes
3answers
272 views

Singularity-free stationary electro-vacuum solution

Let's say we have a spherically symmetric fluid: $$ T^{\alpha \beta} = \begin{bmatrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 ...
3
votes
0answers
345 views

I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) ...
16
votes
1answer
1k views

Flux through a Mobius strip

I was sent here from mathoverflow, hoping for a complete answer to this: === A friend of mine asked me what is the flux of the electric field (or any vector field like $$ \vec r=(x,y,z)\mapsto ...
11
votes
1answer
477 views

Formulation of Transformation optics using a Material Manifold

Dear Community, recently, Transformation optics celebrates some sort of scientific revival due to its (possible) applications for cloaking, see e.g. Broadband Invisibility by Non-Euclidean Cloaking ...