# Geometric representation of two-forms.

• AuraCrystal
In summary, the conversation discusses the concept of two forms and their geometric representation. The wedge product is used to represent the value of a two form, while the cross product is only applicable in three dimensions. The wedge product is a generalization of the cross product and allows for the calculation of "volumes" for integration in higher dimensions. However, the wedge product is not the same as the cross product and they are related through a function. The conversation also touches on the use of the egg crate visual aid and the relationship between the wedge product and the honey-comb analogy.

#### 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.

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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 $dx \wedge dy$.

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

AuraCrystal said:
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

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

AuraCrystal said:
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.

^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?)

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

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

AuraCrystal said:
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.

## 1. What is a two-form?

A two-form is a mathematical object that is used to describe geometric properties of a surface or space. It is a type of differential form that assigns a value to each point on the surface or space.

## 2. How are two-forms represented geometrically?

Two-forms are represented geometrically using vector fields and differential forms. Vector fields represent the direction and magnitude of a vector at each point on the surface or space, while differential forms describe the rate of change of a function in different directions.

## 3. What is the significance of two-forms in geometry?

Two-forms play a crucial role in understanding the curvature and topology of a surface or space. They are used to measure and analyze geometric properties such as area, volume, and angles, making them essential in various fields of mathematics and physics.

## 4. How are two-forms calculated?

Two-forms are calculated using the exterior derivative, which is a mathematical operation that takes a differential form and produces a new one. The exterior derivative is used to differentiate functions in different directions, allowing for the calculation of two-forms.

## 5. In what applications are two-forms commonly used?

Two-forms are commonly used in differential geometry, topology, and mathematical physics. They are also applied in engineering and computer graphics to model and analyze geometric structures and surfaces.