8.1.2 Tensor Indexed Component Scope

Tensor Indexed Component Scope defines how tensor components are indexed and scoped in algebraic structures.

Tensor Indexed Component Scope is the extent, within a specific numerical computation, over which an index attached to a tensor's components is treated as ranging over all its possible values versus being fixed to one particular concrete value. It refines the general notion of index scope to the level of actual numbers: at the component level, an index's scope determines whether that index is still symbolically ranging — as a free or dummy index does — or whether it has been specialized to a fixed integer, in which case it no longer participates in any summation or component-labeling role.

Ranging Indices Versus Fixed Indices

An Index in Ranging Scope

An index in ranging scope is one that has not yet been assigned a specific numerical value: it either labels a free slot of the tensor, standing for "the component at this position, for any valid value of the index," or it is a dummy index being summed over its entire range under the summation convention. In both cases, the index's scope spans the full set 1, ..., n.

An Index Fixed to a Single Value

An index is said to have exited ranging scope, and to be fixed, once a specific numerical value has been substituted for it, as in evaluating T^{2}_{3} from the general component T^{i}_{j}. A fixed index no longer contributes to any implicit sum and no longer labels a family of components; it picks out exactly one number from the tensor's component array.

T_{j}^{i} \underset{fix i=2, j=3}{\mapsto} T_{3}^{2}

Scope Boundaries Within a Single Expression

Partial Fixing

An expression may fix some of a tensor's indices while leaving others in ranging scope, as in T^{i}_{3}, where j has been fixed to the value 3 while i remains free. The component scope of such an expression is mixed: it describes an entire family of numbers, one for each value of i, all drawn from the single fixed value of j.

Component Scope Under Contraction

When a dummy index is summed, its component scope — the full range 1 through n — is exhausted by the summation itself: every one of its possible values contributes a term to the sum, and once the sum is carried out, that index no longer appears, having been entirely absorbed within its own scope.

T^{i} v_{i} = T^{1} v_{1} + T^{2} v_{2} + ... + T^{n} v_{n}

Diagram of Component Scope

Interaction with Free-Index Consistency

Free Indices Retain Ranging Scope Across Terms

When a tensor equation carries a free index, that index must remain in the same ranging scope on every term of the equation: the equation is really shorthand for n separate scalar equations, one for each value the free index may take, and no term may fix that index while another leaves it ranging, since the two sides would then no longer describe the same family of components.

Dummy Indices Cannot Leak Their Scope Outward

Once a dummy index has been summed, its scope is closed; it cannot be referenced again outside the term in which it was bound, since after summation there is no longer any single value of that index left to point to — the sum has already combined all of them.

Practical Consequences

Numerical Evaluation Requires Exiting Ranging Scope

Producing an actual number from a tensor expression — as opposed to a symbolic formula describing a family of components — requires that every index eventually be fixed: free indices are fixed by choosing which component of the result is wanted, and dummy indices are fixed one at a time internally as the sum is carried out term by term.

Implementation in Computational Settings

In code that manipulates tensor components as multidimensional arrays, this distinction corresponds to the difference between an index used as a loop variable, whose component scope spans the array's full extent, and an index used as a fixed array offset, whose scope has collapsed to a single entry; conflating the two is a common source of indexing errors in tensor computations carried out numerically.