7.18.1 Tensor Component Index Range

Tensor Component Index Range is the set of whole number values that a given index of a tensor is permitted to take, determined by the dimension of the vector space on which the tensor is defined, and it fixes how many distinct positions that index contributes to when the tensor's components are enumerated.

Determining the Range

Dependence on the Dimension of the Space

For a tensor defined on a vector space of dimension n, each index of the tensor independently ranges over the whole numbers from 1 to n, or equivalently from 0 to n minus 1 depending on the indexing convention in use. The Tensor Component Index Range for any single index is therefore the set:

with the same range applying uniformly to every index of the tensor, regardless of whether that index is contravariant or covariant, since variance type affects how an index transforms rather than how many values it may take.

Consistency Across All Indices of a Tensor

Every index belonging to a single tensor shares the same Tensor Component Index Range, because all indices of a tensor are associated with the same underlying vector space and its dual. A tensor defined on a three-dimensional space has every one of its indices ranging over exactly three values, whether the tensor has one index, two indices, or many more.

Illustration

The marked points represent the individual values available to a single index. The Tensor Component Index Range is the complete set of these marked points, bounded below by the first value and above by the dimension of the space, n.

Effect on the Total Number of Components

Multiplying Ranges Across Multiple Indices

For a tensor with several indices, each ranging independently over its own Tensor Component Index Range, the total number of components, before any symmetry or antisymmetry is taken into account, is the product of the sizes of the ranges for every index. For a rank-r tensor defined on an n-dimensional space, with every index sharing the same range of size n, the total number of components before reduction is n raised to the power r.

Interaction With Symmetry Reductions

The Tensor Component Index Range determines the raw count of positions available to each index before any reduction from symmetric or antisymmetric behavior is applied. Reductions such as the Tensor Component Symmetric Reduction and the Tensor Component Antisymmetric Reduction are computed by taking this raw range as their starting point and then removing redundant or vanishing positions according to the relevant equality or sign-change rule.

Variation Across Different Tensors

Range Depends Only on the Space, Not on the Tensor's Rank or Type

The Tensor Component Index Range of a given index is fixed entirely by the dimension of the vector space involved, and it does not depend on how many other indices the tensor possesses or on whether those other indices are covariant or contravariant. A vector, a rank-two tensor, and a rank-four tensor defined on the same three-dimensional space all have indices ranging over the identical set of three values.

Different Ranges for Different Spaces

If a tensor is instead defined on a vector space of a different dimension, every index of that tensor ranges over a correspondingly different set of values. A tensor defined on a four-dimensional space has indices ranging over four values rather than three, changing the total number of components accordingly.

Relationship to Other Tensor Concepts

Tensor Component Index Range provides the foundational quantity used throughout Tensor Component Enumeration, since counting the components of a tensor, with or without symmetry considerations, begins from knowing how many values each index may independently take. It underlies the raw component counts referenced in both the Tensor Component Symmetric Reduction and the Tensor Component Antisymmetric Reduction as the starting point from which those reduced counts are derived.