194 Two-Form
An antisymmetric multilinear mapping on ordered pairs of vectors that is used to form integrals over two-dimensional domains.
definition (2-form) A bilinear, antisymmetric, real-valued function of two vector inputs, \(\boldsymbol{\omega}: \mathbf{V} \times \mathbf{V} \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:
- (Bilinearity) \(\boldsymbol{\omega}(a\mathbf{u} + b\mathbf{u}', \mathbf{v}) = a \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) + b \cdot \boldsymbol{\omega}(\mathbf{u}', \mathbf{v})\) and \(\boldsymbol{\omega}(\mathbf{u}, a\mathbf{v} + b\mathbf{v}') = a \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) + b \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}')\).
- (Antisymmetry) \(\boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) = -\boldsymbol{\omega}(\mathbf{v}, \mathbf{u})\).
where
- \(\mathbf{V}\) is a real vector space.
- \(\mathbb{R}\) is the set of real numbers.
- \(\boldsymbol{\omega}\) is a 2-form.
- \(\mathbf{u}, \mathbf{u}', \mathbf{v}, \mathbf{v}'\) are vectors in \(\mathbf{V}\).
- \(a, b\) are scalars.
194.1 Elementary Example
194.1.1 Simple
A two-form is antisymmetric on ordered pairs of vectors.
\[ \omega : V \times V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \omega(e_{1},e_{2}) = 1,\quad \omega(e_{2},e_{1}) = -1,\quad \omega(e_{1},e_{1}) = 0 \]
where
- \(\omega\) is the two-form.
194.1.2 General
On \(\mathbb{R}^{3}\), a two-form is a skew-symmetric \(3 \times 3\) matrix of components.
\[ (\omega_{ij}) = \begin{pmatrix} 0 & 1 & 2 \\ -1 & 0 & 3 \\ -2 & -3 & 0 \end{pmatrix} \]
\[ \omega(u,v) = \sum_{i,j} \omega_{ij}\, u^{i}\, v^{j},\quad \omega_{ij} = -\omega_{ji} \]
where
- \(\omega_{ij}\) are the components of \(\omega\).
- Alternating Function
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