168 Forms
An alternating mapping on a vector space that is used to produce a scalar from ordered lists of vectors.
Forms are functions that take vectors as input and return scalars. A 1-form takes a single vector as input, a 2-form takes two vectors as input, and so on.
definition (1-form = linear functional = covector = covariant vector) A linear, real-valued function of a single vector input, \(\boldsymbol{\omega}: \mathbf{V} \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:
- (Additivity) \(\boldsymbol{\omega}(\mathbf{u} + \mathbf{v}) = \boldsymbol{\omega}(\mathbf{u}) + \boldsymbol{\omega}(\mathbf{v})\).
- (Homogeneity) \(\boldsymbol{\omega}(k\mathbf{v}) = k \cdot \boldsymbol{\omega}(\mathbf{v})\).
where
- \(\mathbf{V}\) is a real vector space.
- \(\mathbb{R}\) is the set of real numbers.
- \(\boldsymbol{\omega}\) is a 1-form.
- \(\mathbf{u}, \mathbf{v}\) are vectors in \(\mathbf{V}\).
- \(k\) is a scalar.
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.
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.
168.1 Elementary Example
168.1.1 Simple
A \(1\)-form is a linear map from vectors to scalars.
\[ \omega : V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \omega(e_{1}) = 2,\quad \omega(e_{2}) = -1,\quad \omega(e_{3}) = 0 \]
where
- \(\omega\) is a \(1\)-form.
- \(V\) is the set of input basis vectors.
168.1.2 General
A \(2\)-form is an antisymmetric bilinear map. In components it is a \(3 \times 3\) skew matrix on \(\mathbb{R}^{3}\).
\[ \omega : \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]
\[ (\omega_{ij}) = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 2 \\ 0 & -2 & 0 \end{pmatrix} \]
\[ \omega(u,v) = \sum_{i,j=1}^{3} \omega_{ij}\, u^{i}\, v^{j},\quad \omega_{ij} = -\omega_{ji} \]
where
- \(\omega_{ij}\) are the components of the \(2\)-form.
- \(\omega_{ij} = -\omega_{ji}\) is antisymmetry.
168.2 References
- Spivak, M. Calculus on Manifolds. — multilinear \(k\)-tensor; alternating when swap of arguments changes sign.
- 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