All Questions
Tagged with electromagnetism differential-forms
11
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 $\Omega$:
$$\int_{t_{0}}^{t_{1}}\left(...
- 1,490
1
vote
2
answers
109
views
Interpreting the cohomology class of the Maxwell tensor.
In the introduction to Bott and Tu, “Differential forms in algebraic topology” there is the motivating example of a stationary point charge in $3$-space. The electromagnetic field $\omega$ is a $2$-...
- 385
0
votes
0
answers
75
views
Oersted's Law in differential form notation
I'm preparing to get chewed up on a midterm and I don't yet know how to write Oersted's Law in differential form notation.
The goal is to convert this
$\int_{\partial S} H \cdot dr = \int_{s} \nabla \...
- 581
6
votes
1
answer
2k
views
Geometric Algebra or Differential Forms for Electromagnetism? [closed]
Electromagnetism (Maxwell's equations) are most often taught using vector calculus.
I have read that both geometric algebra and differential forms are ways to simplify the material.
What are some ...
- 7,042
1
vote
1
answer
129
views
What is the correspondence between gauge field terminology and bundle terminology in electromagnetism?
In electromagnetism, the electromagnetic field tensor can be expressed as $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$
If we let $A= A_\mu dx^\mu$, since $F= \frac{1}{2} F_{\mu \nu} dx^\...
user955291
2
votes
1
answer
76
views
Variational derivation of *all* covariant Maxwell's equations?
If I suppose there exists a 4-"vector potential" $A\in\Omega^1(U)$ such that the Faraday 2-form satisfies $F = dA$ (which is equivalent to assuming the homogeneous Maxwell's equations $dF=0$ ...
- 7,532
1
vote
1
answer
616
views
Why does the electromagnetic tensor in component form coincide with the differential-geometric definition of a $2$-form?
From physics classes, I understand the electromagnetic field strength tensor to be defined as
$$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu \;,$$
where $\partial^\mu$ is the partial derivative (...
- 157
9
votes
2
answers
881
views
Biot-Savart law on a torus?
Background: In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $...
- 2,101
1
vote
1
answer
190
views
Electric field of a charge uniformly distributed on a plane
I am supposed to calculate the electric field $E$ created by a electrical charge $Q>0$ distributed on the surface of a plane. For this I should use
(i) Gauss' theorem $$\int_M \operatorname{div}(...
- 2,085
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
1
vote
0
answers
285
views
area of arbitrary surface element
I am a physics student with a minimal background in differential geometry and I am trying to determine an area element on an arbitrary surface. Suppose we have a surface parameterized by a function $z=...
- 377