Background:
I have now studied up on differential forms.
Here are some outcomes of the study
- Understand $\omega:T_p \mathbb{R^n} \rightarrow \mathbb{R}$
- Can integrate m-forms
- Can do derivatives of forms
- Understand Hodge
I have several electromagnetism courses under my belt(not bragging just giving background). My count is 3. ( soon to be 4)
- I can write maxwell's equations in terms of forms
Arena
The transverse gauge (Coulomb gauge).
Going from the gauge condition $\nabla \cdot A = 0$ to writing an expression for A. On a good day, I can do this using physical arguments and basic PDE theory.
Vocabulary: I have not read up on closed forms but it seems they are forms whose derivative is zero, and an exact form is one whose derivative is another form.
#Question
What is the simplest way to state Poincare Lemma such that I can apply it computationally to the coulomb gauge?
I can for example now almost recite the lemma but I don't understand what it means. I think it goes something like "~every closed differential form is locally exact" I don't know how this says anything about the vector potential A.
Below is my old attempt at asking the question. I have left it here for historical purposes.
I am trying to understand gauge fixing.
I want to believe that the coulomb gauge is defined by the gauge condition: $\nabla \cdot A = 0$
Now all I have to do is try to solve for A.
It seems the standard physics trick is to start by making a connection with the magnetic field $B$ in terms of the vector potential. So $B = \nabla \times A$. From here one substitutes $B = \nabla \times A$ into $\nabla \times B = \mu J$ to get a Poisson vector potential.
The next trick is to make physical arguments about sources (solving by greens function) and then write down the standard solution.
$A = \frac{\mu}{4 \pi} \int \int \int \frac{J}{|r-r'|} dV$
How is this done to arrive at the standard expression that one gets?
My understanding is that there is a more direct way this is done in mathematics starting with $\nabla \cdot A =0$ there are some lemmas: like Poincare lemma that magically take you to the standard solution directly. For example, see the application to Electromagnetism in this Wikipedia page The only problem is that I've looked at this lemma for a while now and I am having some trouble with it.