Geometric representation of two-forms.

[AuraCrystal]

I've been browsing through MTW recently and I found something that puzzles me:

They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface containing those two is \mathbf{u} \bigwedge \mathbf{v} (I don't get this, why isn't it just the vector product \mathbf{u} \times \mathbf{v}?) and the value of the two form is just the number of tubes the "egg-crate" structure cuts through this parallelogram. I don't get this.

They also state that the a basis two-form, say \mathbf{d}x \bigwedge \mathbf{d}y can be represented by just crossing the surfaces of each basis one-form. This is also confusing.

 

[Phrak]

The cross product only works in three dimensions. The cross product of vectors u and v is actuall an axial vector, or pseudovector, rather than a true vector. But in four dimensions, which direction would the cross product point?

I think for, some of us, it's better not to use the egg crate visual aid but to just to think of dx \wedge dy as an infinitessimal area element--or you may eventually prefer some combination of the two.

The egg crate cells with 'vortices' in each cell does help visualizing Stoke's theorem.

Writing out both vectors in two dimension,

\mathbf{u} = u_x dx + u_y dy
\mathbf{v} = v_x dx + v_y dy

\mathbf{u} \wedge \mathbf{v} = u_x v_y dx \wedge dy - u_y v_x dx \wedge dy
= (u_x v_y - u_y v_x) dx \wedge dy

This is the parallelogram multiplied by the area infinitessimal dx \wedge dy.

 

[AuraCrystal]

^OK, sooo how does that relate to the differential forms..? (Is it that it's just a function of those infinitesimal areas?)

 

[atyy]

They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface containing those two is \mathbf{u} \bigwedge \mathbf{v} (I don't get this, why isn't it just the vector product \mathbf{u} \times \mathbf{v}?)

The wedge product of two one-forms is an area. In 3D and if a metric is present, it is related to the length of a vector given by the cross-product (a special case of "Hodge duality"). I'm not sure I got that right, see http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_DGeometry2011-1.pdf 5.0.1 Example: The Cross product in R³

 

[AuraCrystal]

So it's in a way the generalization of the cross product?

 

[atyy]

So it's in a way the generalization of the cross product?

Yes. The wedge product allows one to define "volumes" for integration without a metric in N-dimensions.

 

[AuraCrystal]

^All right! ^_^

How does it allow you to "cross" the surfaces of the one-forms?

(i.e. dx wedge dy; how does that lead to the honey-comb?)

 

[atyy]

I'm not familiar with this analogy, but I do have MTW. What page is it on?

 

[AuraCrystal]

^It's in chapter 4, pages 99-101.

 

[Phrak]

How does it allow you to "cross" the surfaces of the one-forms?

(i.e. dx wedge dy; how does that lead to the honey-comb?)

It doesn't. The wedge product is not a cross product. They are similar but not the same. Specifically, in R3 for a cross product w(u,v) and a wedge product z(u,v),

w_i = u_j \times v_k = u_j v_k - u_k v_j
z_{jk} = u_j v_k - u_k v_j
where i,j,k are cyclic permutations of 1,2,3.

Notice that the tensor entries are similar. There exists a function that relates the wedge product to the cross product.