193 Tensors

A multilinear mapping on vectors and covectors that assigns a scalar that is used to represent quantities whose values transform consistently under a change of basis.

definition [d] (Tensor = Multilinear Map) A multilinear, real-valued function of type \((r,s)\) on a vector space \(V\) and its dual \(V^{*}\):

\(T : \underbrace{V^{*} \times \cdots \times V^{*}}_{r} \times \underbrace{V \times \cdots \times V}_{s} \rightarrow \mathbb{R}\) .

where

\(T\) is the tensor.
\(V\) is a vector space over \(\mathbb{R}\).
\(V^{*}\) is the dual space of \(V\).
\(r\) is the contravariant rank.
\(s\) is the covariant rank.
\(\mathbb{R}\) is the set of real numbers.

Note:

\(V\) may equally be a vector space over \(\mathbb{C}\).
\(T\) is linear in each argument independently.
a scalar is type \((0,0)\); a vector is type \((1,0)\); a covector is type \((0,1)\).

definition [d] (Tensor) An array of components \(T^{i_1\ldots i_p}_{\ j_1\ldots j_q}\) labeled by \(p\) contravariant upper and \(q\) covariant lower indices, whose components transform linearly under a coordinate change \(x \mapsto x'\):

(Contravariant) \((T')^{i} = \displaystyle \sum_{j} \dfrac{\partial x^{j}}{\partial (x')^{i}}\, T^{j}\) .
(Covariant) \((T')_{i} = \displaystyle \sum_{j} \dfrac{\partial (x')^{i}}{\partial x^{j}}\, T_{j}\) .
(Mixed rank 2) \((T')^{i}_{\ j} = \displaystyle \sum_{k,l} \dfrac{\partial x^{k}}{\partial (x')^{i}}\, \dfrac{\partial (x')^{j}}{\partial x^{l}}\, T^{k}_{\ l}\) .

where

\(T^{i_1\ldots i_p}_{\ j_1\ldots j_q}\) are the components of the tensor.
\(p\) is the number of contravariant indices.
\(q\) is the number of covariant indices.
\(x\) and \(x'\) are the old and new coordinates.
\(T^{j}\), \(T_{j}\), \(T^{k}_{\ l}\) are components in the old coordinates.
\((T')^{i}\), \((T')_{i}\), \((T')^{i}_{\ j}\) are components in the new coordinates.

Note:

the total rank is \(p + q\); in \(d\) dimensions a rank-\(n\) tensor has \(d^{n}\) components.
upper indices transform with the Jacobian \(\partial x^{j}/\partial (x')^{i}\); lower indices with its inverse \(\partial (x')^{i}/\partial x^{j}\).
the transformation laws ensure the tensor represents a coordinate-independent geometric object.

193.1 Examples

example 1 [d] (Euclidean Metric on \(\mathbb{R}^{3}\) — Frankel, Arfken) On the fixed vector space \(V = \mathbb{R}^{3}\) with the standard basis, the Euclidean inner product is the type-\((0,2)\) tensor

\(g : V \times V \rightarrow \mathbb{R}\) ,
\(g(u, v) = u \cdot v = \sum_{i,j} \delta_{ij}\, u^{i}\, v^{j}\) ,

with components

\((g_{ij}) = \operatorname{diag}(1,\, 1,\, 1) = I_{3}\) .

As a multilinear map this is one bilinear form on \(V\), not a field of forms on a manifold.

where

\(V = \mathbb{R}^{3}\) is the underlying vector space.
\(u, v \in V\) are vectors.
\(u^{i}, v^{j}\) are components of \(u\) and \(v\) in the standard basis.
\(\delta_{ij}\) is the Kronecker delta.
\(I_{3}\) is the \(3 \times 3\) identity matrix.

Note:

this is a single tensor of type \((0,2)\) on \(V\).
contrast with a metric tensor field, whose components \(g_{ij}(x)\) depend on the point.

example 2 [d] (Mixed Kronecker Tensor — Carroll, Arfken) On a finite-dimensional vector space \(V\), the identity map \(\operatorname{id}_{V} : V \rightarrow V\) is the type-\((1,1)\) tensor \(\delta\) with components

\(\delta^{i}_{\ j} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\) ,

so that

\(\delta(v, \omega) = \omega(v)\) ,
\(\delta^{i}_{\ j}\, v^{j} = v^{i}\) .

where

\(V\) is a finite-dimensional vector space.
\(\delta^{i}_{\ j}\) are the components of the mixed Kronecker tensor.
\(v \in V\) is a vector with components \(v^{j}\).
\(\omega \in V^{*}\) is a covector.

Note:

\(\delta^{\mu}_{\ \nu}\) is the identity map on vectors and one-forms.
the components are numerical constants, independent of any point on a manifold.

example 3 [d] (Minkowski Metric Components — Griffiths) As an array of components, the flat spacetime metric tensor is

\((g_{\mu\nu}) = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\) .

where

\(g_{\mu\nu}\) are the components of the Minkowski metric tensor.
\(\mu, \nu\) are spacetime indices running over \(0,1,2,3\).

Note:

this is a type-\((0,2)\) tensor written as a \(4 \times 4\) array of components.
the same array is often written \(\eta_{\mu\nu} = \operatorname{diag}(-1,\, 1,\, 1,\, 1)\).

193.2 Elementary Example

193.2.1 Simple

A type-\((0,2)\) tensor returns a scalar from two vectors. On two basis vectors, list the four values.

\[ T : V \times V \rightarrow \mathbb{R} \]

\[ V = \{ e_{1},\ e_{2} \} \]

\[ T(e_{1},e_{1}) = 2,\quad T(e_{1},e_{2}) = 1,\quad T(e_{2},e_{1}) = 1,\quad T(e_{2},e_{2}) = 3 \]

193.2.2 General

In three dimensions a type-\((0,2)\) tensor is a \(3 \times 3\) matrix of components \(T_{ij}\).

\[ T : \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]

\[ (T_{ij}) = \begin{pmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{pmatrix} \]

Let \(g\) be the Euclidean metric tensor on \(\mathbb{R}^{3}\). It is the type-\((0,2)\) tensor whose components form the \(3 \times 3\) identity matrix.

\[ g : \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]

\[ (g_{ij}) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I_{3} \]

\[ g(u,v) = \sum_{i,j=1}^{3} g_{ij}\, u^{i}\, v^{j} = u \cdot v \]

where

\(g\) is the Euclidean metric tensor.
\(g_{ij}\) are its components in the standard basis.
\(u^{i}, v^{j}\) are components of vectors \(u, v\) in \(\mathbb{R}^{3}\).
\(I_{3}\) is the \(3 \times 3\) identity matrix.

Let \(\delta\) be the mixed Kronecker tensor. It is the type-\((1,1)\) tensor for the identity map, with components \(\delta^{i}_{\ j}\).

\[ \delta : (\mathbb{R}^{3})^{*} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]

\[ (\delta^{i}_{\ j}) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

\[ \delta^{i}_{\ j}\, v^{j} = v^{i} \]

where

\(\delta\) is the mixed Kronecker tensor.
\(\delta^{i}_{\ j}\) equals \(1\) if \(i = j\) and equals \(0\) if \(i \neq j\).
\(v^{j}\) are components of a vector \(v\) in \(\mathbb{R}^{3}\).

193.3 Historical Notes

Tensor calculus was established by Ricci and Levi-Cività in 1901.[8] Einstein adopted tensors for general relativity to ensure physical laws—such as the unified stress-energy-momentum tensor—remain invariant across all coordinate systems.[8]

193.4 References

Nash, C., & Sen, S. Topology and Geometry for Physicists. Academic Press, 1983. — tensor as multilinear map.
Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — multilinear-map definition; mixed Kronecker as type \((1,1)\).
Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — component transformation laws; Kronecker delta; Euclidean metric components.
Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — component transformation definition.
Cahill, K. Physical Mathematics. Cambridge University Press, 2019. — component transformation definition.
Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — Euclidean metric as bilinear form.
Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — Minkowski metric as component matrix \(g_{\mu\nu}\).
Gowers, T., Barrow-Green, J., & Leader, I. (Eds.). The Princeton Companion to Mathematics. Princeton University Press, 2008. — Part VIII.7 chronology (p. 1013): Ricci and Levi-Cività, 1901, Méthodes du Calcul Différentiel Absolut et leurs Applications; Part IV.13 Dafermos, general covariance and the stress–energy–momentum tensor \(T\).

192 Tensor Field

194 Two-Form