Two-vector

A two-vector or bivector^[1] is a tensor of type $\scriptstyle {\binom {2}{0}}$ and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).

The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then

\mathbf {f} =f^{\alpha \beta }\,{\vec {e}}_{\alpha }\otimes {\vec {e}}_{\beta }

where the f ^{α β} are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition.

An example of a bivector is the stress–energy tensor.

References

^ Penrose, Roger (2004). The road to reality : a complete guide to the laws of the universe. New York: Random House, Inc. pp. 443–444. ISBN 978-0-679-77631-4.

[1] Penrose, Roger (2004). The road to reality : a complete guide to the laws of the universe. New York: Random House, Inc. pp. 443–444. ISBN 978-0-679-77631-4.

[1]

See also

References