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2021-01-02T07:28:24Z

Task 18 (cosmetic): eval 2 templates: del empty params (4×);

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A '''two-vector''' or '''bivector'''<ref>{{cite book |last=Penrose |first=Roger |year=2004 |title=The road to reality : a complete guide to the laws of the universe |location=New York |publisher=Random House, Inc. |pages=443–444 |isbn=978-0-679-77631-4}} Note: This book mentions "bivectors" (but not "two-vectors") in the sense of <math>\scriptstyle\binom{2}{0}</math> tensors.</ref> is a [[tensor]] of type <math>\scriptstyle\binom{2}{0}</math> and it is the [[dual space|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 [[Vector (geometric)|vector]]s 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<ref>{{cite book |last=Schutz |first=Bernard |year=1985 |title=A first course in general relativity |location=Cambridge, UK |publisher=Cambridge University Press |page=77 |isbn=0-521-27703-5}} Note: This book does not appear to mention "two-vectors" or "bivectors", only <math>\scriptstyle\binom{2}{0}</math> tensors. </ref>
:<math> \mathbf{f} = f^{\alpha \beta} \, \vec e_\alpha \otimes \vec e_\beta </math>
where the ''f <sup>α β</sup>'' are the components of the two-vector. Notice that both indices of the components are [[Covariance and contravariance of vectors|contravariant]]. This is always the case for two-vectors, by definition. A bivector may operate on a one-form, yielding a vector:
:<math> f^{\alpha \beta} u_{\beta} = v^{\alpha}</math>,
although a problem might be which of the upper indices of the bivector to contract with. (This problem does not arise with mixed tensors because only one of such tensor's indices is upper.) However, if the bivector is [[symmetric tensor|symmetric]] then the choice of index to contract with is indifferent.

An example of a bivector is the [[stress–energy tensor]]. Another one is the ''orthogonal complement''<ref>Penrose, op. cit., §18.3</ref> of the [[metric tensor]].
 
==Matrix notation==
If one assumes that vectors may only be represented as column matrices and covectors as row matrices; then, since a square matrix operating on a column vector must yield a column vector, it follows that square matrices can only represent mixed tensors. However, there is nothing in the [[abstract algebra|abstract algebraic]] definition of a matrix that says that such assumptions must be made. Then dropping that assumption matrices can be used to represent bivectors as well as two-forms. Example:

<math>\begin{pmatrix}f^{00} && f^{01} && f^{02} && f^{03} \\
f^{10} && f^{11} && f^{12} && f^{13} \\
f^{20} && f^{21} && f^{22} && f^{23} \\
f^{30} && f^{31} && f^{32} && f^{33} \end{pmatrix} \begin{pmatrix}u_0 \\ u_1 \\ u_2 \\ u_3\end{pmatrix} =
\begin{pmatrix}f^{00} u_0 + f^{01} u_1 + f^{02} u_2 + f^{03} u_3\\
f^{10} u_0 + f^{11} u_1 + f^{12} u_2 + f^{13} u_3\\
f^{20} u_0 + f^{21} u_1 + f^{22} u_2 + f^{23} u_3\\
f^{30} u_0 + f^{31} u_1 + f^{32} u_2 + f^{33} u_3\end{pmatrix} = \begin{pmatrix}v^0 \\ v^1 \\ v^2 \\ v^3\end{pmatrix}
\iff f^{\alpha \beta} u_\beta = v^\alpha</math>

<math>\begin{pmatrix}u_0 && u_1 && u_2 && u_3\end{pmatrix} \begin{pmatrix}f^{00} && f^{01} && f^{02} && f^{03} \\
f^{10} && f^{11} && f^{12} && f^{13} \\
f^{20} && f^{21} && f^{22} && f^{23} \\
f^{30} && f^{31} && f^{32} && f^{33} \end{pmatrix}</math>

<math>
= \begin{pmatrix}u_0 f^{00} + u_1 f^{10} + u_2 f^{20} + u_3 f^{30} &&
u_0 f^{01} + u_1 f^{11} + u_2 f^{21} + u_3 f^{31} &&
u_0 f^{02} + u_1 f^{12} + u_2 f^{22} + u_3 f^{32} &&
u_0 f^{03} + u_1 f^{13} + u_2 f^{23} + u_3 f^{33}\end{pmatrix}</math>

<math> = \begin{pmatrix} w^0 && w^1 && w^2 && w^3\end{pmatrix} \iff u_\alpha f^{\alpha \beta} = f^{\alpha \beta} u_\alpha = w^\beta</math> or <math>f^{\beta \alpha} u_\beta = w^\alpha</math>.

If ''f'' is symmetric, i.e., <math>f^{\alpha \beta} = f^{\beta \alpha}</math>, then <math>v^\alpha = w^\alpha</math>.

==See also==
* [[Two-point tensor]]
* [[Bivector#Tensors and matrices|Bivector § Tensors and matrices]] (but note that the stress–energy tensor is symmetric, not skew-symmetric)
* [[Dyadics]]

==References==
<references />

{{DEFAULTSORT:Two-Vector}}
[[Category:Tensors]]

Two-vector - Revision history

imported>Monkbot: Task 18 (cosmetic): eval 2 templates: del empty params (4×);