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> For a concrete example, one youtuber showed how Maxwells equations simplified to a single equation if you introduce an operator that is a combination of div a
by rsj_hn 5y ago
> For a concrete example, one youtuber showed how Maxwells equations simplified to a single equation if you introduce an operator that is a combination of div and curl, and also a new kind of physical entity that combines the electrical and magnetic fields.
Back when I was in university, we covered this in our differential geometry class. And yes, you'd use more abstract concepts like curvature, hodge dual, and exterior product.
Maxwells equations in any dimension can be reduced to:
dF = 0 and d*F = 0
That's two equations, not one, but you can introduce a new D = (d, d*) and then get DF=0 if you want.
The advantage here is the d, and F have all the old physical meanings. F is curvature, which is the electro-magnetic field E+B, and d is the derivative (exterior derivative, but that is the derivative needed in calculus).
Here is a derivation:
http://home.lu.lv/~sd20008/papers/essays/Maxwell's%20equations.pdf http://home.lu.lv/~sd20008/papers/essays/Maxwell's%20equatio...
- Grustaf 5y agoThat looks pretty much equivalent, and they even have a version where F = E + B, just like in the youtube video. But my question is if this F, or the tensor version for that matter, has any physical meaning?
- rsj_hn 5y agoYes, F is the electromagnetic field whose laws of motion generate the E/M dynamics. https://en.wikipedia.org/wiki/Electromagnetic_tensor https://en.wikipedia.org/wiki/Electromagnetic_tensor (This assumes you believe a "Field" has a physical meaning.)
- Grustaf 5y agoIt’s pretty easy to think about a normal three dimensional vector field representing the magnetic field, the trouble for me is when you combine the two into one tensor field. Looking at the link reminds me that this entity is used outside of GA, but it still feels a bit weird. More weird than say complex currents, but maybe it actually isn’t.
- jabl 5y agoI think the point is that when learning EM, it's a lot about developing intuition by visualizing in your head how the E and B fields behave in different situations, and how charges interact with them etc. You know, a lot of holding your right hand in the "physicist handshake" position and twisting it. Sure, the same information is contained in the F tensor, but it doesn't have a similar straightforward geometric intuition.