8.1.4 Tensor Indexed Summation Scope

Tensor Indexed Summation Scope is the precise boundary of a single implicit sum generated by the Einstein summation convention: the exact set of multiplicative factors within a tensor expression over which a repeated index — occurring once as a superscript and once as a subscript — is understood to be summed, before that sum is treated as a completed quantity for any further operation applied to it. It is the narrowest of the scope notions used in tensor index notation, confined to identifying where one particular implicit sum begins and ends.

What Delimits a Summation's Scope

Multiplicative Adjacency Within a Term

The summation convention links a repeated index across factors that are multiplied together within a single term; the scope of that summation is exactly the product of factors carrying the repeated index, and it extends no further than the boundary of that multiplicative term, marked by the nearest addition, subtraction, or other non-multiplicative operation.

Here the summation scope of i covers only A^{i}B_{i}, since C^{k} lies in the same multiplicative term but does not share the letter i; the addition sign marks the end of that term, so D_{k} — despite sharing the letter k with C^{k} — lies in a wholly separate summation scope on the other side of the plus sign, and in fact k here is a free index, appearing once in each of the two terms, rather than a dummy index summed within either.

Parentheses as Explicit Scope Delimiters

Where a summation scope might otherwise be ambiguous, parentheses make the boundary explicit: in (A^{i}_{j} B^{j}_{k}) C^{k}_{l}, the parentheses confine the summation scope of j strictly to the enclosed product A^{i}_{j} B^{j}_{k}, with the outer contraction over k treated as a separate summation scope applied only after the inner one has been fully evaluated.

Nested Summation Scopes

One Sum Feeding Into Another

When a contracted quantity from one summation scope is itself contracted again with another tensor, the two summations form nested scopes: the inner scope must be completed — conceptually, though not necessarily in actual sequential computation — before the result participates in the outer sum, mirroring the order of operations used for ordinary nested arithmetic sums.

Multiple Independent Sums Within One Term

A single term can contain more than one distinct summation scope simultaneously, one for each independently repeated index pair, as in T^{ij}_{k} S^{k}_{ij}, where i, j, and k are each summed, but the three summation scopes all coincide with the same overall term, since all three pairs are multiplicatively adjacent within it.

Diagram of Summation Scope Boundaries

Summation Scope Versus Equation Scope and Symbol Scope

Narrower Than Equation Scope

Summation scope governs only the local completion of one implicit sum, whereas equation scope governs the broader requirement that free indices agree across an entire equation; a dummy index's summation scope always lies strictly within the equation scope of the equation containing it, never spanning across the equals sign or across separate additive terms.

Independent of Symbol Scope

The document-wide convention reserving certain letters for certain roles — symbol scope — has no bearing on where any one summation's scope begins or ends; a letter conventionally reserved for spatial indices throughout a chapter still has its summation scope determined purely by the local multiplicative structure of the specific term in which it appears.

Practical Importance of Identifying Summation Scope Correctly

Avoiding Miscounted Sums

Misjudging the boundary of a summation scope — for instance, treating an index as summed across an addition sign rather than confined to one term — produces an expression that does not correspond to any valid tensor operation and generally fails to satisfy the free-index balance required of a well-formed tensor equation.

A Prerequisite for Mechanical Manipulation

Symbolic and computational systems that manipulate tensor expressions rely on correctly parsing summation scope to decide which factors must be looped over together when evaluating a contraction; an incorrect determination of scope in such a system would combine unrelated factors into a single erroneous sum or fail to combine factors that should be summed together.