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.