#bivector

[Click here for a PDF version of this post] Motivation. I was asked about geometric algebra equivalents for a couple identities found in [1], one for line integrals \begin{equation}\label{eqn:more_feynmans_trick:20} \ddt{} \int_{C(t)} \Bf \cdot d\Bx = \int_{C(t)} \lr{ \PD{t}{\Bf} + \spacegrad \lr{ \Bv \cdot \Bf } – \Bv \cross \lr{ \spacegrad \cross \Bf } } \cdot d\Bx, \end{equation} and one for…

[Click here for a PDF version of this post] On LinkedIn, James asked for ideas about how to solve What is the total area of ABC? You should be able to solve this! using geometric algebra. I found one way, but suspect it’s not the easiest way to solve the problem. To start with I’ve re-sketched the triangle with the areas slightly more to scale in fig. 1, where areas $A_1, A_2, A_3, A_5$ are…

[Click here for a PDF version of this and previous two posts] The author of a book draft I am reading pointed out if a vector transforms as \begin{equation}\label{eqn:adjoint:760} \Bv \rightarrow M \Bv, \end{equation} then cross products must transform as \begin{equation}\label{eqn:adjoint:780} \Ba \cross \Bb \rightarrow \lr{ \textrm{adj}\, M }^\T \lr{ \Ba \cross \Bb }. \end{equation} Bivectors…

[Click here for the PDF version of this post.] Motivation. This revisits my last blog post where I covered this content in a meandering fashion. This is an attempt to re-express this in a more compact form. In particular, in a form that is amenable to include in my book. When I wrote the potential section of my book, I cheated, and didn’t try to motivate the results. My cheat was figuring out the…

[Click here for a PDF version of this post] On discord, on the bivector server, ‘not a good username’ asked a question that I really liked. It’s a question that nagged me before too, but I hadn’t taken the time to puzzle through it properly. The main character in this question is the spherical polar form of a radial vector, which has the…

[Click here for a PDF version of this post] Conventional formulation. The idea behind introducing the scalar potential \( \phi \) and vector potential \( \BA \) is that we can impose a constraint on the form of our observable fields \( \BE, \BB \), (or \( \BD, \BH \)), that reduces the complexity and coupling of Maxwell’s equations. These potentials are not unique, but the types of allowed…

New video (on Google’s CensorshipTube): Click here to watch the video on the Odysee platform. [Click here for a PDF version of this post (and supplementary notes for the video.)] Look below for a wordpress version of this post (and supplementary notes for the video.) Exact system. Recall that we can use the wedge product to solve linear systems. For example, assuming that \( \Ba, \Bb \) are not…

[Click here for a PDF version of this post] There’s a class of simple statics problems that are pervasive in high school physics and first year engineering classes (for me that CIV102.)  These problems are illustrated in the figures below. Here we have a static load under gravity, and two supporting members (rigid beams or wire lines), which can be under compression, or tension, depending on the…