#grade selection

[Click here for a PDF version of this post] This is a continuation of yesterday’s post on the relationships between the exterior derivative, and the curl operation (grad-wedge) in geometric algebra. Hodge star vs. pseudoscalar multiplication. We find a definition of the hodge star for basic k-forms in [2]. Definition 1.7: Hodge star. Let \( \omega \) be a basic k-form on \(\mathbb{R}^n\). The…

A better 3D generalization of the Mandelbrot set.

I’ve been exploring 3D generalizations of the Mandelbrot set: Visualizing the 3D Mandelbrot set, and Some 3D renderings of the Mandelbrot set, Slicing of the 3D Mandelbrot set, and analysis. The iterative equation for the Mandelbrot set can be written in vector form ([1]) as: \begin{equation} \begin{aligned} \Bz &\rightarrow \Bz \Be_1 \Bz + \Bc \\ &= \Bz \lr{ \Be_1 \cdot \Bz } + \Bz \cdot \lr{…

Potential solutions to the static Maxwell’s equation using geometric algebra

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When neither the electromagnetic field strength \( F = \BE + I \eta \BH \), nor current \( J = \eta (c \rho – \BJ) + I(c\rho_m – \BM) \) is a function of time, then the geometric algebra form of Maxwell’s equations is the first order multivector (gradient) equation \begin{equation}\label{eqn:staticPotentials:20} \spacegrad F = J. \end{equat…

Reciprocity theorem in Geometric Algebra

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The reciprocity theorem involves a Poynting like antisymmetric difference of the following form

\begin{equation}\label{eqn:reciprocityTheoremGA:20} \BE^{(a)} \cross \BH^{(b)} – \BE^{(b)} \cross \BH^{(a)}. \end{equation}

This smells like something that can probably be related to a combined electromagnetic field multivectors in some sort of…