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Electromagnetism using geometric algebra versus components

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Re: Electromagnetism using geometric algebra versus components

#11

Post by DreamScatter »

yiyus wrote: post #9
aesthesia wrote: post #6

Earlier quoted context omitted.

You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.

This "competition" between geometric algebra and differential forms makes me uncomfortable. As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily don…

You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra

The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.

Re: Electromagnetism using geometric algebra versus components

#12

Post by enriquto »

Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument.

The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).

Re: Electromagnetism using geometric algebra versus components

#14

Post by yiyus »

DreamScatter wrote: post #11
yiyus wrote: post #9

Earlier quoted context omitted.

This "competition" between geometric algebra and differential forms makes me uncomfortable. As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily don…

You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.

You can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would you then say that geometric algebra occurs in differential forms?

In any case, you did not attempt to answer my original question. Are GA and DFs just different ways to define "equivalent" concepts or is there some more fundamental difference that I am missing?

Re: Electromagnetism using geometric algebra versus components

#15

Post by carapace »

I found "Geometric Algebra for Electrical and Electronic Engineers" helpful: https://ieeexplore.ieee.org/document/6876131

ABSTRACT: In this paper, we explicate the suggested benefits of Clifford’s geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.

Re: Electromagnetism using geometric algebra versus components

#16

Post by DreamScatter »

yiyus wrote: post #14
DreamScatter wrote: post #11

Earlier quoted context omitted.

You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.

You can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would y…

No, I would say differential forms occur in geometric algebra, not the other way around.

Re: Electromagnetism using geometric algebra versus components

#17

Post by creata »

yiyus wrote: post #9
aesthesia wrote: post #6

Earlier quoted context omitted.

You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.

This "competition" between geometric algebra and differential forms makes me uncomfortable. As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily don…

If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate bilinear form to define the Hodge star, from which you can just as easily define the geometric product, and from the geometric product the Hodge star.

To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalent. Every geometric algebra gives rise to a Hodge star, and an exterior algebra, and so on, but exterior algebras are a much more general concept, so they're less powerful until you tack that extra structure on.

Re: Electromagnetism using geometric algebra versus components

#18

Post by jbay808 »

bawana wrote: post #13

If a point is the intersection of two lines, and a line is the intersection of two planes, then what structures intersect to give a plane? Two 3D spaces? And so the intersection of 2 4D spaces should give a 3D space? How do I express these ideas with GA?

Total non expert here but your fascinating question got me thinking out loud.

Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but...

Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle".

So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.

Re: Electromagnetism using geometric algebra versus components

#19

Post by wwarner »

sgdpk wrote: post #2

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 v…

+1 to this exposition!

Re: Electromagnetism using geometric algebra versus components

#20

Post by orbots »

yiyus wrote: post #9
aesthesia wrote: post #6

Earlier quoted context omitted.

You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.

This "competition" between geometric algebra and differential forms makes me uncomfortable. As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily don…

There is a fairly nuanced difference that doesn't really matter much unless you are a mathematician. Essentially in GA you'd do your differential geometry assuming a sort of ambient background space. In regular differential geometry the space of forms and vectors are abstracted and don't require a shared geometric embedding. I'm not a mathematician, so this is a very non-precise explanation, but that's how I understand it.
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