1.2.55 Tensor Component Definition

Tensor components are numerical values that represent parts of a tensor in a specific coordinate system, defined through its basis and index structure.

Tensor Component Definition is the specification of a tensor through the numerical values it takes when expressed relative to a chosen basis of the underlying vector space and dual space, together with the index structure and transformation behavior that make those numbers represent a single, basis-independent object. While a tensor itself is an abstract multilinear map or an element of a tensor product space, its components are the coordinates obtained once a basis is fixed, arranged in an array indexed by as many labels as the tensor has contravariant and covariant slots.

Components Relative to a Basis

Definition via Basis Expansion

Given a vector space V with basis e_1, ..., e_n and dual basis e^1, ..., e^n of V*, any type (p, q) tensor T can be written as a linear combination of basis tensor products formed from p copies of the e_i vectors and q copies of the e^j covectors. The coefficients in this expansion are the components of T.

T = T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

where the Einstein summation convention applies, meaning repeated upper and lower indices are summed over all values from 1 to n, the dimension of V.

Components as Evaluations of the Multilinear Map

Equivalently, if T is regarded as a multilinear map taking p covectors and q vectors as input, the components are obtained by evaluating T on the basis elements directly:

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = T (e^{i^{1}}, \dots, e^{i^{p}}, e_{j_{1}}, \dots, e_{j_{q}})

This shows that each component is a single scalar produced by feeding the appropriate basis vectors and dual basis covectors into the multilinear map.

Index Placement and Meaning

Upper Indices

Upper indices, written as superscripts such as i_1, ..., i_p, correspond to the contravariant slots of the tensor. Each upper index ranges over the dimension n of V and identifies which coefficient of the basis expansion in the V direction is being referenced.

Lower Indices

Lower indices, written as subscripts such as j_1, ..., j_q, correspond to the covariant slots of the tensor. Each lower index ranges over the same dimension n and identifies which coefficient of the basis expansion in the V* direction is being referenced.

Total Number of Components

A tensor of type (p, q) in an n-dimensional space has n^(p+q) components, since each of the p + q indices independently ranges over n values.

number of components = n^{p + q}

Components Under a Change of Basis

Why Components Are Not Basis-Independent

The numerical values of a tensor's components depend on the choice of basis; the same abstract tensor has different components in different bases, just as the coordinates of a vector change when the coordinate axes are changed. What remains invariant is the tensor itself, not the specific array of numbers representing it.

Transformation Rule

If a new basis is related to the old one by a matrix A, the components transform using A once for each upper index and the inverse matrix A^{-1} once for each lower index:

{\tilde{T}}_{l_{1} \dots l_{q}}^{k^{1} \dots k^{p}} = A_{i_{1}}^{k^{1}} \dots A_{i_{p}}^{k^{p}} {(A^{- 1})}_{l_{1}}^{j^{1}} \dots {(A^{- 1})}_{l_{q}}^{j^{q}} T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

This transformation law is what distinguishes tensor components from an arbitrary array of numbers: the array must transform in exactly this prescribed way under every admissible change of basis for it to represent a genuine tensor.

Examples of Component Arrays

Scalar Components

A type (0, 0) tensor has a single component, a single number that does not change under any change of basis, since there are no indices to transform.

Vector Components

A type (1, 0) tensor has n components, one upper index T^i, and these transform exactly as the coordinates of a vector do.

Covector Components

A type (0, 1) tensor has n components, one lower index T_j, and these transform exactly as the coordinates of a linear functional do.

Matrix-Like Components

A type (1, 1) tensor has n^2 components T^i_j, which can be arranged in a square array and interpreted as the matrix of a linear transformation relative to the chosen basis.

Bilinear Form Components

A type (0, 2) tensor has n^2 components T_{jk}, which can be arranged in a square array representing a bilinear form, such as a metric tensor, relative to the chosen basis.

Diagrammatic Summary

The diagram highlights that the component array T carries a fixed set of upper indices, drawn from the contravariant slots, and lower indices, drawn from the covariant slots, with each index independently ranging over the dimension of the vector space.