13.21.1 Tensor Contraction Summation Notation

Tensor Contraction Summation Notation simplifies tensor operations by summing over repeated indices, key in physics and engineering.

Tensor Contraction Summation Notation is the notational form in which the summation implied by a contraction is written out using an explicit summation symbol and a stated range for the summed index, offered as an alternative to the implicit repeated index notation when clarity, unfamiliarity with the Einstein convention, or explicit statement of the summation range is preferred.

Definition

Summation notation writes a contraction as an explicit sum:

\sum_{a = 1}^{n} A^{a} B_{a}

explicitly displaying both the summation symbol and the range $a = 1$ to $n$ , rather than relying on the reader to infer the summation from the mere repetition of the index symbol.

Relationship to Repeated Index Notation

Equivalence in Meaning

Summation notation and repeated index notation denote exactly the same quantity; the two differ only in whether the summation is displayed explicitly or left implicit:

\sum_{a} A^{a} B_{a} = A^{a} B_{a}

When Explicit Notation Is Preferred

Explicit summation notation is preferred in contexts where the underlying convention linking repetition to summation cannot be assumed, where the summation range is non-standard or needs to be emphasized, or where a mixture of summed and non-summed repeated symbols could otherwise create ambiguity.

Components of the Notation

Summation Symbol

The symbol $\sum$ marks the location of an explicit sum, distinguishing the operation from ordinary multiplication of the terms that follow it.

Range Specification

The range attached to the summation symbol, typically written beneath and, when needed, above it, states precisely which values the summed index takes, removing any ambiguity about the dimension of the space being summed over.

Summand

The expression following the summation symbol is the summand, evaluated once for each value of the summed index within its stated range, with the results added together to form the total.

Diagram

Position Within Notation

Summation notation serves as the explicit counterpart to repeated index notation within the broader system of tensor contraction notation, and the two forms are used interchangeably depending on context, with the choice between them affecting only the visual presentation of an expression and never its underlying mathematical meaning or the value it denotes.