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    <title>Electromagnetism using Geometric Algebra versus Components</title>
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    <link>http://www.av8n.com/physics/maxwell-ga.htm</link>
    <dc:creator>nico.ash</dc:creator><description><![CDATA[The task for today is to compare some more-sophisticated and less-sophisticated ways of expressing the laws of electromagnetism. In particular we compare Geometric Algebra, ordinary vectors, and vector components.

We do this in the spirit of the correspondence principle: whenever you learn a new formalism, you should check that it is consistent with what you already know.

    This document is also available in PDF format. You may find this advantageous if your browser has trouble displaying standard HTML math symbols. ]]></description>
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    <title>Electromagnetism using geometric algebra versus components | Hacker News</title>
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    <link>https://news.ycombinator.com/item?id=25219503</link>
    <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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    <dc:creator>nico.ash</dc:creator><description><![CDATA[Illustrated Self-Guided Course On
How To Use The Slide Rule]]></description>
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<item rdf:about="http://hackershelf.com/">
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<item rdf:about="http://www.bathsheba.com/sculpt/">
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<item rdf:about="http://ideolalia.com/">
    <title>ideolalia - is it strange to dance so soon</title>
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    <title>SciPy – the embarrassing way to code</title>
    <dc:date>2009-11-11T10:32:44+00:00</dc:date>
    <link>http://www.vetta.org/2008/05/scipy-the-embarrassing-way-to-code/?</link>
    <dc:creator>nico.ash</dc:creator><description><![CDATA[The downside to all this is that you send months developing your complex algorithms and when you’re done you show somebody the result of all your efforts — a page or two of code.  It looks like something that somebody could have written in an afternoon.  Even worse, you start to suspect that if you had really known scipy and spent a few days carefully thinking about the problem to start with, then you probably could have coded it in an afternoon.  It’s a little embarrassing.
]]></description>
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    <title>Hacker News | Shazam: not magic after all</title>
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    <link>http://news.ycombinator.com/item?id=909263</link>
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<item rdf:about="http://www.frank-buss.de/lisp/functional.html">
    <title>Functional Geometry</title>
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    <link>http://www.frank-buss.de/lisp/functional.html</link>
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<item rdf:about="http://www.georgehart.com/slide-togethers/slide-togethers.html">
    <title>Slide-Together Geometric Constructions</title>
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<item rdf:about="http://home.in.tum.de/~lehmanna/lisp-tutorial.html">
    <title>Alexander Lehmann</title>
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    <link>http://home.in.tum.de/~lehmanna/lisp-tutorial.html</link>
    <dc:creator>nico.ash</dc:creator><description><![CDATA[writing a simple raytracer in common lisp
]]></description>
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<item rdf:about="http://pcg.wikidot.com/">
    <title>Procedural Content Generation: Main Page</title>
    <dc:date>2008-05-18T20:43:00+00:00</dc:date>
    <link>http://pcg.wikidot.com/</link>
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</item>
<item rdf:about="http://freakonomics.blogs.nytimes.com/2008/04/18/phil-gordon-answers-your-poker-questions/">
    <title>Phil Gordon Answers Your Poker Questions - Freakonomics - Opinion - New York Times Blog</title>
    <dc:date>2008-04-19T00:13:52+00:00</dc:date>
    <link>http://freakonomics.blogs.nytimes.com/2008/04/18/phil-gordon-answers-your-poker-questions/</link>
    <dc:creator>nico.ash</dc:creator><dc:subject>poker economics psychology math</dc:subject>
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</item>
<item rdf:about="http://www.theatlantic.com/doc/200204/rauch">
    <title>Seeing Around Corners</title>
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    <link>http://www.theatlantic.com/doc/200204/rauch</link>
    <dc:creator>nico.ash</dc:creator><description><![CDATA[using artificial socirty to get a peek at society and possible outcomes of policy
]]></description>
<dc:subject>culture math ai simulation economics sociology psychology society emergence science</dc:subject>
<dc:identifier>https://pinboard.in/u:nico.ash/b:6cbd240089bc/</dc:identifier>
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<item rdf:about="http://bldgblog.blogspot.com/2007/12/adventures-in-stacking.html">
    <title>BLDGBLOG: Adventures in Stacking</title>
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    <link>http://bldgblog.blogspot.com/2007/12/adventures-in-stacking.html</link>
    <dc:creator>nico.ash</dc:creator><description><![CDATA[New Scientist published an awesome little article this week about nothing more complex than stacking blocks of wood (subscriber-only)... But, oh, how complex that task can be.
It's the combinatorial architecture of the well-balanced stack.
]]></description>
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