1.2.58 Tensor Component Value Definition

Tensor component values are numerical entries in a tensor that represent its magnitude and direction in multi-dimensional space.

Tensor Component Value Definition is the specification of the actual numerical entry, an element of the underlying field, that a tensor component takes on once every one of its indices has been assigned a fixed integer value between 1 and n, where n is the dimension of the vector space. Where the notion of an index describes the labeling scheme used to address a component, and the component array describes the full collection of all such entries, the component value is the concrete number sitting at one particular address within that array.

From Indices to a Single Value

Fixing All Indices

A tensor of type (p, q) has components written with p upper indices and q lower indices, each index ranging independently from 1 to n. Assigning specific integers to every one of the p + q indices selects exactly one entry from the component array, and that entry is the component value at that particular index assignment.

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} evaluated at i^{1} = a_{1}, \dots, j_{q} = b_{q}

produces a single element of the field F, the component value at that combination of index assignments.

The Value Is an Element of the Field

Regardless of the rank of the tensor or the dimension of the vector space, every individual component value is a single element of the field over which the vector space is defined, most commonly the real numbers or the complex numbers. This is true even for a rank-two or higher tensor: although the array as a whole has many indices, each single entry within it is just one number.

Component Values and Basis Dependence

Values Change with the Basis

The specific numerical value assigned to a component at a fixed index combination depends on the basis chosen for the vector space. The same abstract tensor, evaluated in two different bases, generally produces different numbers at corresponding index positions, even though both arrays describe the same underlying object.

Values Follow the Transformation Law Collectively

While an individual component value has no meaning independent of the chosen basis, the full set of component values as a whole obeys a strict transformation law when the basis changes, ensuring that the entries computed in a new basis can always be recovered from the entries computed in the old basis together with the change-of-basis matrix and its inverse.

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

The individual value on the left is computed as a specific combination of individual values on the right, weighted by entries of the transformation matrices.

Special Values

The Zero Component Value

A component value may equal zero, indicating that the tensor has no contribution along the particular combination of basis directions selected by that index assignment. A tensor whose every component value is zero, in every index combination, is called the zero tensor of that type.

Diagonal Component Values

For rank-two tensors displayed as a matrix, the component values with equal index assignments, such as T^i_i for a fixed i, are called the diagonal values. In special tensors such as the identity map or certain metric tensors, these diagonal values carry particular significance, for instance equaling one in the standard basis representation of the identity.

Symmetric and Antisymmetric Value Relations

If a tensor is symmetric in two indices, the component value at one ordering of those indices equals the component value at the swapped ordering. If a tensor is antisymmetric in two indices, the component value at one ordering equals the negative of the component value at the swapped ordering, which forces every diagonal value in those two indices to equal zero.

T_{i i} = - T_{i i} \Rightarrow T_{i i} = 0

Diagrammatic Summary

The diagram shows that fixing the index assignment i = 2, j = 3 within a rank-two component array picks out one specific entry, T23, whose numerical value is the tensor component value at that index combination.