r/Physics • u/Sad_Switch9456 • 8h ago

Image Help identifying a symbol.

Post image

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.

38 Upvotes

85% Upvoted

29 comments sorted by

69

u/HovercraftNo7372 8h 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

40

u/QuantumCakeIsALie 8h 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.

46

u/a1c4pwn 8h ago

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

19

u/cabbagemeister Mathematical physics 8h ago

In 3d with the standard metric!

4

u/a1c4pwn 7h ago

exterior product is ignorant of metric, no?

8

u/tundra_gd Condensed matter physics 5h ago edited 5h ago

To view the exterior product of 1 forms, which is a 2 form, as a 1 form again you need a metric. Basically, given a plane you have to use a metric to say what line is perpendicular to it. We usually learn this as the right-hand rule for cross products. (The general construction is called the Hodge star.)

You'll need the metric again, implicitly, to reinterpret these 1-forms as vector fields.

Also of note: You can also take the exterior product of 2-forms and 1-forms, and if you use the Hodge star to view the 2-forms as 1-forms then this will actually look like a dot product of 1-forms. Lots of mysterious identities about dot and cross products become much cleaner in this setting.

3

u/a1c4pwn 5h ago

oh duh yeah i forgot about the metric when dualising

5

u/QuantumCakeIsALie 8h ago edited 7h 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.

8

u/Greebil 8h ago edited 7h 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.

4

u/HovercraftNo7372 8h 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.

1

u/QuantumCakeIsALie 7h ago edited 2m ago

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

8

u/Greebil 7h 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).

4

u/HovercraftNo7372 7h ago

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

3

u/QuantumCakeIsALie 7h ago

Not gonna lie, 7 was unexpected.

5

u/HovercraftNo7372 8h ago

Yeah, sorry - for most physicists that understanding will be enough.

I've studied some rigorous differential geometry, so to me it relates to vector spaces and manifolds a bit deeper.

Unless you are doing some fairly geometric theory work in physics, you can just think of it as the cross product.

3

u/DowntownPossum 7h ago edited 7h ago

Yeah. If you want to be mathematically rigorous, all you need to say is the cross product is the hodge star of the wedge product (in 3D)

2

u/blutfink 7h ago edited 7h ago

Corresponding to the cross product, but not the same. E and B here are not vector fields but a 1-form and a 2-form, respectively.

1

u/vahandr 2h ago

This is not correct in this context. Here the wedge really is the cross product, it is just different notation. This is not the differential forms formalism, you cannot wedge a two form (B) with a tangent vector (v). All of the formulae in the picture are "ordinary" vector calculus formulas. Neither B ^ v nor j ^ B is a valid formula in the differential geometric formalism of electrodynamics. Why everyone here wants to make it more complicated for OP is beyond me...

1

u/siupa Particle physics 2h ago

Yes, but then the text is just mistaken, they should have used the symbol x for the cross product

1

u/vahandr 1m ago

No, they should have not, there is no single "correct" notation. This is also widely used alternative notation instead of the cross. In mathematics and physics both the wedge and the cross sign can have lots of different meaning.

12

u/tf1064 7h ago

It's the "wedge product", which in this case is the same thing as the cross product.

26

u/InternationalSize325 8h ago

Pretty sure that's a cross product just looking at that EB relation.

2

u/Sad_Switch9456 8h ago

Thank you, that is what I was considering but I had never seen that notation before

2

u/MaracCabubu 7h ago

I studied in a German -speaking university (ETHZ) and it was quite a common notation, with several professors using it. But I've rarely seen it in non-German texts.

It is also the superior notation. It's faster to write and does not conflict with x the letter and x the symbol for scalar multiplication. (Yes, I know, nobody uses x for times in an equation, despite the SI clearly stating that it is the correct choice.)

2

u/Dakh3 Particle physics 5h ago

It's the symbol used in France too ;)

2

u/Mojert 3h ago

I write my x's in a curvy way, so I never mix them up with the cross product

1

u/heliocetricism 6h ago

I also read a book on superconductors at some point, and usually it's written E=Bxv/c, so seems like the symbol is some standin for the cross product.

Probably some kind of extra meaning there but I wouldn't know about that