185 One-Form
A linear functional on a vector space that is used as the integrand of a line integral.
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.
185.1 Elementary Example
185.1.1 Simple
A one-form is a linear functional on vectors.
\[ \omega : V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \omega(e_{1}) = 3,\quad \omega(e_{2}) = -1,\quad \omega(e_{3}) = 4 \]
where
- \(\omega\) is the one-form.
- \(e_{i}\) are basis vectors.
185.1.2 General
On \(\mathbb{R}^{3}\), \(\omega\) is a row of components acting by contraction on a column of vector components.
\[ \omega(v) = \omega_{1} v^{1} + \omega_{2} v^{2} + \omega_{3} v^{3} \]
\[ (\omega_{1},\omega_{2},\omega_{3}) = (3,-1,4) \]
where
- \(\omega_{i}\) are the components of the one-form.
- 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