174 k-form
An alternating multilinear function of k vectors that is used to measure oriented size of a k-dimensional subspace.
definition (k-form) A multilinear, completely antisymmetric, real-valued function of \(k\) vector inputs, \(\boldsymbol{\omega}: \mathbf{V}^k \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:
(k-Linearity) \(\boldsymbol{\omega}(\mathbf{v}_1, \dots, a\mathbf{v}_i + b\mathbf{v}_i', \dots, \mathbf{v}_k) = a \cdot \boldsymbol{\omega}(\mathbf{v}_1, \dots, \mathbf{v}_i, \dots, \mathbf{v}_k) + b \cdot \boldsymbol{\omega}(\mathbf{v}_1, \dots, \mathbf{v}_i', \dots, \mathbf{v}_k)\) for each argument \(i\) where
\(1 \leq i \leq k\).
(Complete Antisymmetry) Swapping any two vector inputs reverses the sign of the output: \(\boldsymbol{\omega}(\dots, \mathbf{u}, \dots, \mathbf{v}, \dots) = -\boldsymbol{\omega}(\dots, \mathbf{v}, \dots, \mathbf{u}, \dots)\).
where
- \(\mathbf{V}\) is a real vector space.
- \(\mathbf{V}^k\) is the \(k\)-fold Cartesian product of \(\mathbf{V}\).
- \(\mathbb{R}\) is the set of real numbers.
- \(\boldsymbol{\omega}\) is a \(k\)-form.
- \(\mathbf{v}_1, \dots, \mathbf{v}_k, \mathbf{v}_i', \mathbf{u}, \mathbf{v}\) are vectors in \(\mathbf{V}\).
- \(k\) is a positive integer representing the degree of the form.
- \(a, b\) are scalars.
174.1 Elementary Example
174.1.1 Simple
A \(2\)-form is an alternating bilinear function of two vectors.
\[ \omega : V \times V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \omega(e_{1},e_{2}) = 2,\quad \omega(e_{2},e_{1}) = -2,\quad \omega(e_{1},e_{1}) = 0 \]
where
- \(\omega\) is a \(k\)-form with \(k = 2\).
174.1.2 General
On \(\mathbb{R}^{3}\), a \(2\)-form has a skew \(3 \times 3\) component matrix.
\[ (\omega_{ij}) = \begin{pmatrix} 0 & 2 & 0 \\ -2 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix} \]
\[ \omega(u,v) = \sum_{i,j} \omega_{ij}\, u^{i}\, v^{j} \]
where
- \(\omega_{ij}\) are the components of the \(2\)-form \(\omega\).
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
- Covariant Rank
- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
- Differential k-Form
- Edge of a Domain
- Exterior Algebra and Exterior Derivatives
- Forms
- Functions
- Inverse Metric
- k-Covector
- k-Covector Field
- k-Fold Product
- k-form
- k-Tensor
- Lie
- Lie Algebra
- Line Element
- Linear Functional
- Lorentzian Manifold
- Metric Tensor
- Multilinear Function
- Nondegenerate
- Nondegenerate Bilinear Form
- One-Form
- Smooth
- Smooth Assignment
- Smooth Mapping
- Stoke’s Theorem and the Fundamental Theorem of Calculus
- Symmetric Array
- Tangent Space
- Tensor Field
- Tensors
- Two-Form
- Type-(0,2) Tensor Field
- Type-(q,r) Tensor