Two-form

From Wikipedia, the free encyclopedia

This article or section is in need of attention from an expert on the subject.
WikiProject Mathematics or the Mathematics Portal may be able to help recruit one.
If a more appropriate WikiProject or portal exists, please adjust this template accordingly.

A two-form is a bilinear form

$\mathbf{f} : V \times V \rightarrow \mathbb{R}$

which maps any pair of vectors belonging to a vector space to a scalar, in such a way that the mapping is invariant with respect to coordinate transformations of the vector space, and such that interchanging the vectors inverts the sign of the scalar. A two-form can be pictured as an oriented surface defined by the two vectors. A two-form is an antisymmetric tensor of type $\begin{pmatrix} 0 \\ 2 \end{pmatrix}$ .

The above definition may be modified by currying, so that a two-form can also be a linear function $\mathbf{f} : V \rightarrow V \rightarrow \mathbb{R}$ or $\mathbf{f} : V \rightarrow \tilde{V}$ which maps any vector of a vector space V to a one-form of the dual space $\tilde{V}$ . Then, when such two-form is supplied with a pair of vector arguments, it takes in the first vector and returns a one-form, which then takes in the second vector and returns a real number, so the net result remains that a two-form reduces a pair of vector arguments into a scalar.

A two-form can also be described as a linear function

$\mathbf{f} : V \otimes V \rightarrow \mathbb{R}$

which maps a two-vector to a scalar.

A pair of one-forms can be combined by means of the tensor product, whose symbol is $\otimes$ , in order to yield a two-form. A tensor $\tilde{f} \otimes \tilde{g}$ is defined as meaning that it is applied to a pair of vectors $\vec u$ and $\vec v$ by the following rule ("mixed product property"):

$(\tilde{f} \otimes \tilde{g}) \, \vec u \ \vec v = (\tilde{f} \otimes \tilde{g}) (\vec u \otimes \vec v) = \tilde{f} (\vec u) \ \tilde{g} (\vec v)$ ,

the right side of which rule is a product of two scalars, each of which scalars is the result of applying a one-form to a vector. Such product is generally not commutative.

The components of the tensor product $\tilde{f} \otimes \tilde{g}$ are

$(\tilde{f} \otimes \tilde{g})_{\alpha \beta} = (\tilde{f} \otimes \tilde{g}) (\vec e_\alpha \otimes \vec e_\beta) = \tilde{f} (\vec e_\alpha) \ \tilde{g} (\vec e_\beta) = f_\alpha \ g_\beta$ ,

that is,

$(\tilde{f} \otimes \tilde{g})_{\alpha \beta} = f_\alpha \ g_\beta$

which, considering the two-form as a matrix, corresponds to the Kronecker product of a row vector and a column vector to produce a matrix.

Any two-form can be expressed as a linear combination of outer products of basis one-forms, with the scalar coefficients being the components of the two-form:

$\mathbf{f} = f_{\alpha \beta} \ \tilde{\omega}^\alpha \wedge \tilde{\omega}^\beta = f_{\alpha \beta} \ \tilde{\omega}^{\alpha \beta}$

where the $\tilde{\omega}^{\alpha \beta}$ are the basis two-forms.

The components f_{α β} of a two-form can be thought of as being arrayed in a square matrix. If

f αβ = f βα

is true for all components of a two-form, then the two-form is said to be symmetric. If, on the other hand,

f αβ = - f βα

is true for all components of a two-form, then the two-form is said to be anti-symmetric or skew-symmetric.

Categories: Pages needing expert attention from Mathematics experts | Differential forms

Two-form

From Wikipedia, the free encyclopedia

[edit] See also

Views

Navigation

interaction

Search