175 k-Tensor
A multilinear function of k vectors that is used to represent quantities independent of a change of basis.
definition (k-Tensor) A multilinear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following condition applies:
- The function \(T\) is linear in each of its \(k\) arguments.
where
- \(V\) is a vector space.
- \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\).
- \(k\) is a positive integer representing the degree of the tensor.
- \(\mathbb{R}\) is the set of real numbers.
- \(\mathcal{L}^k(V)\) is the vector space of all \(k\)-tensors on \(V\).
Note:
- \(\mathcal{L}^k(V)\) is also written \(\mathcal{J}^k(V)\).
175.1 Elementary Example
175.1.1 Simple
A \(2\)-tensor is a bilinear real-valued map on two vectors. It need not be alternating.
\[ T : V \times V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2} \} \]
\[ T(e_{1},e_{2}) = 4,\quad T(e_{2},e_{1}) = 1,\quad T(e_{1},e_{1}) = 0 \]
where
- \(T\) is a \(k\)-tensor with \(k = 2\).
175.1.2 General
On \(\mathbb{R}^{3}\), the same \(2\)-tensor is a general \(3 \times 3\) matrix of components \(T_{ij}\).
\[ (T_{ij}) = \begin{pmatrix} 0 & 4 & 0 \\ 1 & 2 & 3 \\ 0 & 5 & 6 \end{pmatrix} \]
\[ T(u,v) = \sum_{i,j=1}^{3} T_{ij}\, u^{i}\, v^{j} \]
where
- \(T_{ij}\) are the components of \(T\) in the standard basis.
- 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