All Questions
Tagged with electromagnetism differential-geometry
21
questions
6
votes
2
answers
131
views
Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$?
In an electrodynamics book I came across the following claim for the electric current density (twisted) 2-form $j$ along the boundary of some 3-dimensional volume $\Omega$:
$$\int_{t_{0}}^{t_{1}}\left(...
- 1,490
1
vote
2
answers
109
views
Interpreting the cohomology class of the Maxwell tensor.
In the introduction to Bott and Tu, “Differential forms in algebraic topology” there is the motivating example of a stationary point charge in $3$-space. The electromagnetic field $\omega$ is a $2$-...
- 385
1
vote
1
answer
110
views
Stokes theorem 2 sides not matching with Magnetic waves
We have been asked to verify stokes theorem for a magnetic field.
We know Stokes theorem states, for any vector field $\vec{H}$:
$$\int_S{(\nabla \times \vec{H}) \cdot \vec{dS}} = \oint_L{\vec{H} \...
- 123
3
votes
0
answers
54
views
Are there theorems in fiber bundle land and differential geometry land that make calculations in electromagnetism easier?
Some time ago while thinking about life and such, I thought to myself does recasting electromagnetism in bundle theory make certain calculations easier? To be precise are there theorems in ...
- 581
2
votes
0
answers
130
views
casting electromagnetism for exterior differential calculus
I am trying to understand curved spacetime Maxwell's equations in terms of exterior differential calculus. I am surfacing this topic due to working with a flat, but non-constant metric and I am ...
- 101
1
vote
0
answers
70
views
Why is $F_{\mu \nu}$ fundamentally geometric in nature?
I am studying gauge theory, and we derive the Lagrangian for electrodynamics by wanting the Lagrangian to be gauge invariant under U(1) symmetry group. That is, invariant under the phase rotation, $$\...
user955291
1
vote
1
answer
871
views
How do we compute Hodge duals?
The motivation for this question is to try to come up with a general expression for $(\star F)_{\mu\nu}$, the $\mu,\nu$ component of the Hodge dual of the Field strength tensor, which is of great ...
- 12.5k
4
votes
1
answer
185
views
Why is the Kaluza-Klein ansatz the natural choice?
In Kaluza-Klein theory we can choose a parametrisation for the 5-dimensional metric: $$d\hat{s}^2 \equiv \hat{g}_{ab} dx^a dx^b = g_{\mu\nu}dx^\mu dx^\nu + \phi^2(dz + A_\mu dx^\mu)^2 $$
where $g_{\mu\...
- 51
1
vote
1
answer
616
views
Why does the electromagnetic tensor in component form coincide with the differential-geometric definition of a $2$-form?
From physics classes, I understand the electromagnetic field strength tensor to be defined as
$$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu \;,$$
where $\partial^\mu$ is the partial derivative (...
- 157
1
vote
0
answers
73
views
Representing flux tubes as a pair of level surfaces in R^3
I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of ...
- 341
2
votes
0
answers
41
views
Show that $\nabla . *F = 0$ is a geometric frame independent of ...
Show that $\nabla\cdot\ast F = 0$ (divergence of the dual of the EMF tensor) is a geometric frame independent version of $F_{ab,c} + F_{bc,a} + F_{ca,b} = 0$, where $F$ is the electromagnetic field ...
- 761
9
votes
2
answers
881
views
Biot-Savart law on a torus?
Background: In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $...
- 2,101
1
vote
0
answers
42
views
Spatial curves that obey $z'-r^2 \theta' = \text{const.}$ in cylindrical coordinates
I am interested in a class of (arc-length parametrized) curves $\gamma:\mathbb{R} \to \mathbb{R}^3$ with the following property:
If the curve is written in cylindral coordinates $(r,\theta,z)$, it ...
- 179
3
votes
0
answers
130
views
Do Maxwell's equations (generalized) apply to _every_ $k$-form on a pseudo-Riemannian manifold?
Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G={\...
- 716
0
votes
0
answers
190
views
Delta distribution and divergence theorem
Let's say we have some vector field $\vec C$ such that $$\operatorname{div}\vec C=-\mu_0\vec j=-\mu_0 I\,\delta(x)\,\delta(y)\,\vec e_z.$$ where $\mu_0$ and $I$ are constants.
I am interested in the ...
- 1,395