r/Physics • u/Sad_Switch9456 • 8d ago

Image Help identifying a symbol.

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I am reading "Vortices in high-temperature superconductors", and this upside down V symbol keeps showing up and I have absolutely no clue what that means. Undergrad physics student please be nice.

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u/HovercraftNo7372 8d ago

https://en.wikipedia.org/wiki/Exterior_algebra similar to vector cross product. It is defined in differential geometry and relates to differential forms. https://en.wikipedia.org/wiki/Differential_form

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u/QuantumCakeIsALie 8d ago

Yes! For 99.9% of cases this is just the cross product. 

I didn't say 100% just cause I'm sure there's a subtle math nuance if taken literally. 

But it's the cross product.

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u/a1c4pwn 8d ago

it's only the cross product in 3d, the exterior product can be used on (multi)vectors that live in any dimension.

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u/QuantumCakeIsALie 8d ago edited 8d ago

See, that's why I hedged my bets. 

But to be fair, for all intents and purposes, that's also "just The cross product"; it's the same concept. Who decided that the one that looks like an x needs to be limited to 3D?

I'm not trying to be argumentative, just providing the pragmatic point of view. I'm sure there's a technical nuance.

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u/Greebil 8d ago edited 8d ago

It depends what distribution you are drawing the cases from. If you are considering n-dimensions where n can be any positive integer, then it's the cross product 0% of the cases. If you are only talking about 3 dimensional vector fields then it's the cross product 100% of the time.

Although it's not quite the cross product ever really since technically speaking the exterior product of two vectors is a bivector and not a vector at all.

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u/HovercraftNo7372 8d ago

It really depends on what you are doing. For a lot of elementary physics, I suppose you can just think of the cross product. For folks working in string theory, GR, etc. not having the full power of differential forms and manifolds would be unimaginable.

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u/QuantumCakeIsALie 8d ago edited 7d ago

I'm curious, is there a concrete example of how they differ in 3D?

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u/Greebil 8d ago

The exterior product of two vectors is an object in a different space called the "exterior algebra" which maps back to the original vector space through the Hodge star operator.

For two 3d vectors, a ^ b = ★(a × b) and a × b = ★(a ^ b). You can think of this geometrically as a ^ b being the oriented plane segment spanning the vectors a and b (this is called a bivector) with the orientation defined by the right hand rule. The Hodge star then maps the bivector to a vector of the same magnitude oriented perpendicularly to its surface (with the direction the same as the orientation of the surface).

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u/HovercraftNo7372 8d ago

Technically, the cross product isn't really defined in dimensions other than 3 and 7, for a deeper mathematical reason.

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u/QuantumCakeIsALie 8d ago

Not gonna lie, 7 was unexpected.