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What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.

by lrasinen5 hours ago|

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And then there are us engineers. I don't care much either way whether Maxwell's equations are ∇F = J or some other form, as long as it makes the problem easier to solve.

If I were in the GA Marketing Committee I'd publish a paper with suitably hand-picked worked examples where the vector approach is long and tedious, and GA version is short and sweet.

by simpaticoder3 hours ago|

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I like this idea but I get the sinking feeling GA proponents don't really solve problems with GA. Like how Haskell advocates don't write programs and modular synth enthusiasts don't write music.

by radialstub3 hours ago|

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Application of the Method of Moments to solve full wave formulation of Maxwell's equations. To derive the EFIE using maxwell's equations is a massive pain. With geometric algebra, all you need is ∇F = J and the MoM becomes a mechanical process.

by eigenspace6 hours ago|

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I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form).

Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.

by jsLavaGoat6 hours ago|

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There isn't just ONE interesting facet of this. There isn't just ONE mathematical formalism of a lot of these things. GA is just one of those approaches and you should see it just once, just like you should see the group structure and all of that as well. For most applications, the standard vector calculus approach is fine. But the math underlying all of this is full of richness and no one approach is the skeleton key.

Same with programming languages. Some people are like RUST RUST RUST and some are like C C C! I'm like, you guys only use one language?

by chombier3 hours ago|

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> pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.

I don't know, I recently tried to work out how the metric on vectors/1-forms induces a metric on higher-degree forms, and if the geometric product magically gives this for free I'd say it's a win (same for the Hodge star).

by eigenspace2 hours ago|

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Both differential forms and geometric algebra are awkward for that sort of thing. I'd just stick with abstract index notation most of the time.

by immmmmm5 hours ago|

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agreed, when you start needing the the hodge star, diff form loose quite a lot of their interest.

i'd add it's quite nice in string theories for RR fields and coupling to D-branes, where writing 10 anti-symmetrized indices quickly gets annoying.. and topological field theories..