Two-form

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A two-form is a bilinear form

[\mathbf : V \times V \rightarrow \mathbb]

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. A two-form is a tensor of type [\begin 0 \\ 2 \end].

The above definition may be modified by currying, so that a two-form can also be a linear function [\mathbf : V \rightarrow V \rightarrow \mathbb ] or [ \mathbf : V \rightarrow \tilde ] which maps any vector of a vector space V to a one-form of the dual space [\tilde]. 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 : V \otimes V \rightarrow \mathbb ]

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 \otimes \tilde] 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 \otimes \tilde) \, \vec u \ \vec v = (\tilde \otimes \tilde) (\vec u \otimes \vec v) = \tilde (\vec u) \ \tilde (\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 \otimes \tilde] are

[(\tilde \otimes \tilde)_ = (\tilde \otimes \tilde) (\vec e_\alpha \otimes \vec e_\beta) = \tilde (\vec e_\alpha) \ \tilde (\vec e_\beta) = f_\alpha \ g_\beta ],

that is,

[(\tilde \otimes \tilde)_ = 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_ \ \tilde^\alpha \wedge \tilde^\beta = f_ \ \tilde^ ]

where the [\tilde^] 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_ = f_ ]

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

[ f_ = -f_ ]

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

Two-form

See also