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    <title>Electromagnetism using geometric algebra versus components | Hacker News</title>
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    <dc:creator>nico.ash</dc:creator><description><![CDATA[ sgdpk 4 hours ago [–]

Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?

With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.

If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.

You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.

[1] https://en.wikipedia.org/wiki/Pseudovector

reply


bollu 4 hours ago [–]

For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)

reply


pastrami_panda 3 hours ago [–]

This is also an excellent and interactive blog-post building up to the understanding of how rotors can replace quaternions in GA: https://marctenbosch.com/quaternions/

reply


patrec 3 hours ago [–]

For clicking convenience: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ (the link above is broken since HN includes the ]; in the URL).

reply ]]></description>
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http://www.spacetime.us/wiki
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It's the combinatorial architecture of the well-balanced stack.
]]></description>
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