alex@810lab.com
Maxwell's Operator¶
Summary¶
We explore an operator related to Maxwell's equations, and try to respect differential geometry.
Idea¶
Consider the following operator :
$$e^{\nabla} = 1+\nabla + \nabla^2 + \ldots$$
Ignore the higher order terms for now. Apply it to the vector potential, $A$.
\begin{align} e^{\nabla}A &= A+\nabla A + \nabla^2 A = A+F + J \end{align}
Which is interesting enough to look more at this operator. Given a definition of the vector derivative in terms of a basis[1],
$$\nabla \equiv \sum e^i \partial_i .$$ where $e^i \equiv e_i^{-1}$, and $\partial_i \equiv \partial_{e_i}$ is the differential in the $e_i$ direction. This implies $$ e^{\nabla} = e^{ \sum e^i \partial_i} \stackrel{?}{=} \prod e^{ e^i \partial_i}$$
Lets reverse the order of logic and assume $\prod e^{ e^i \partial_i}$ is true. Then under what circumstance does the equality hold: $$ \prod e^{ e^i \partial_i} \stackrel{?}{=} e^{ \sum e^i \partial_i}=e^{\nabla} $$ expanded in a cartesian frame in STA, $$ e^{ e^0 \partial_0}e^{ e^1 \partial_1}e^{ e^2 \partial_2}e^{ e^3 \partial_3} \stackrel{?}{=} e^{ e^0 \partial_0 + e^1 \partial_1+ e^2 \partial_2+ e^3 \partial_3} $$
This is almost identical to the transmission line model used in [2]. In the transmission line case, the equality was justified by the fact that the rotations were small (and in null bivectors). However, the rotations relative magnitude is what produced the range of behavoir.
Transmission line Model¶
\begin{align} R &\equiv e_{30} − e_{13} \\ X &\equiv e_{12} − e_{20} \\ G &\equiv e_{30} + e_{13} \\ B &\equiv e_{12} + e_{20} \end{align}
Others¶
How can split this up by div/curl:
\begin{align} e^{\nabla}\cdot A &= A + J \\ e^{\nabla}\wedge A &= F \end{align}
References¶
[1] Hestenes, D. (1984). Clifford Algebra to Geometric Calculus: A Practical Language for Physics. D. Reidel Publishing Company.
[2] A. Arsenovic, "Applications of Conformal Geometric Algebra to Transmission Line Theory," in IEEE Access, vol. 5, pp. 19920-19941, 2017, doi: 10.1109/ACCESS.2017.2727819.