Potential Formulation via Differential Forms#
Warning
This is a very first draft
In this page, I derive the potential formulation of the equations of Mr. Maxwell via differential form. This article in its current form is a first draft and may contain errors.
Inhomogenous Maxwell equations via differential forms
\[\begin{split}d\:⋆ \left[ \begin{aligned}
- & \E^x \; dt ∧ dx \\
- & \E^y \; dt ∧ dy \\
- & \E^z \; dt ∧ dz \\
& B^x \; dy ∧ dz \\
& B^y \; dz ∧ dx \\
& B^z \; dx ∧ dy \\
\end{aligned} \right]
= \begin{bmatrix}
+ μ_0 c ρ \; dx ∧ dy ∧ dz\\
- μ_0 J^x \; dt ∧ dy ∧ dz\\
- μ_0 J^y \; dt ∧ dz ∧ dx\\
- μ_0 J^z \; dt ∧ dx ∧ dy\\
\end{bmatrix}\end{split}\]
Homogenous Maxwell equations via differential forms
\[\begin{split}d \left[ \begin{aligned}
- & \E^x \; dt ∧ dx \\
- & \E^y \; dt ∧ dy \\
- & \E^z \; dt ∧ dz \\
& B^x \; dy ∧ dz \\
& B^y \; dz ∧ dx \\
& B^z \; dx ∧ dy \\
\end{aligned} \right]
= 0\end{split}\]
\[d F^{♭♭} = 0\]
Potential formulation
Taking the exterior derivative of any differential form \(A\) twice is zero
\[ddA = 0\]
Since \(dA = F^{♭♭}\), we can infer that \(A\) is a one-form \(A=A^{♭}\). From the Homogenous equations we obtain:
\[\begin{split}d\:d \left[ \begin{alignedat}{2}
- & φ & dt \\
& A^x & dx \\
& A^y & dy \\
& A^z & dz \\
\end{alignedat} \right]
= 0\end{split}\]
Now from the inhomogenous equations:
\[\begin{split}d\:⋆ d\left[ \begin{alignedat}{2}
- & φ \; & dt \\
& A^x \; & dx \\
& A^y \; & dy \\
& A^z \; & dz \\
\end{alignedat} \right]
= \begin{bmatrix}
+ μ_0 c ρ \; dx ∧ dy ∧ dz \\
- μ_0 J^x \; dt ∧ dy ∧ dz \\
- μ_0 J^y \; dt ∧ dz ∧ dx \\
- μ_0 J^z \; dt ∧ dx ∧ dy \\
\end{bmatrix}\end{split}\]
Maxwell Equations are in short form:
\[d ⋆ d A^♭ = J^{♭♭♭}\]
As expected, we can add a term do A of the form \(dα\) since this is guaranteed to be zero and \(α\) can be any form:
\[\begin{split}α = number + \left[ \begin{aligned}
α^t \; dt \\
α^x \; dx \\
α^y \; dy \\
α^z \; dz \\
\end{aligned} \right]
+ \left[ \begin{aligned}
α^? \; d? ∧ d? \\
α^? \; d? ∧ d? \\
α^? \; d? ∧ d? \\
α^? \; d? ∧ d? \\
\end{aligned} \right]
+ \left[ \begin{aligned}
α^? \; dx ∧ dy ∧ dz \\
α^? \; dt ∧ dy ∧ dz \\
α^? \; dt ∧ dz ∧ dx \\
α^? \; dt ∧ dx ∧ dy \\
\end{aligned} \right]\end{split}\]