8.11.3 Tensor Explicit Summation Term

The Tensor Explicit Summation Term expresses the summation of tensor components using indices, providing a clear and direct representation of tensor operations in algebra.

Tensor Explicit Summation Term is the individual summand expression that follows an explicit sigma symbol, representing the specific quantity that is computed once for each admissible value of the summation index and then added together with every other such computed value to produce the total value of the sum.

Role of the Term Within the Sum

One Evaluation Per Index Value

The explicit summation term is evaluated once for every value the summation index takes across its stated range, with the index appearing inside the term substituted by each concrete value in turn before the resulting numbers are added together.

\sum_{i = 1}^{3} A_{i} B^{i} = A_{1} B^{1} + A_{2} B^{2} + A_{3} B^{3}

Here the term (A_i B^i) is the pattern evaluated three separate times, once for each value (i) takes from one to three, producing the three individual products that are then added together.

Terms Following an Explicit Sigma Are Determined by Scope

The extent of the summation term, that is, exactly which part of the written expression is understood to be governed by a given sigma symbol, is determined by the notational scope of that symbol, and correctly identifying where the term ends and any subsequent, unrelated expression begins is essential to reading the notation correctly.

Terms Involving Free Indices

Free Indices Persisting Through Every Instance of the Term

When the summation term also contains a free index, in addition to the summation index bound by the sigma, that free index remains present, with the same value, throughout every individual evaluation of the term across the range of the bound index.

\sum_{i = 1}^{n} A_{i k} B^{i}

In this term, (i) is bound and swept across its range by the sigma, while (k) remains fixed at whatever value is under consideration for that particular component of the result, unaffected by the summation.

Terms in Nested Summations

Inner and Outer Terms

When a summation term itself contains a nested sigma symbol, the innermost sigma's term is fully evaluated and summed first, for each fixed value of the outer index, before the outer summation adds together the resulting values.

\sum_{i = 1}^{n} (\sum_{j = 1}^{n} A_{i j})

Relationship to Einstein Notation

The Term Persists, the Symbol Vanishes

When an explicit summation is rewritten in Einstein convention notation, the summation term itself is exactly what remains written on the page; only the sigma symbol and the explicitly stated range are dropped, since the term's own repeated, opposite-variance index is sufficient to signal that the same summation is intended.

\sum_{i = 1}^{n} A_{i} B^{i} \to A_{i} B^{i}

Practical Illustration

Understanding the explicit summation term as the reusable pattern that is evaluated once per index value and then accumulated is the conceptual bridge between the compact notation used to write a sum and the concrete sequence of arithmetic operations that sum actually represents once fully expanded.