8.6.1 Tensor Free Index Occurrence Pattern

Understanding how free indices appear in tensor expressions and their significance in algebraic structures.

Tensor Free Index Occurrence Pattern is the structural regularity that governs how many times, and in what variance, a free index must appear across the terms of a tensor expression in order for that expression to be considered well formed. The pattern requires that every free index appear exactly once per term, that its variance, upper or lower, remain fixed, and that it recur identically across every additive term of the expression.

Core Rule of the Pattern

Single Occurrence Per Term

Within any single term of a tensor expression, a free index occurs exactly once, unpaired with a matching index of opposite variance. This distinguishes it from a dummy index, which occurs exactly twice within a term, once upper and once lower.

F_{i} = M_{i j} X^{j}

In this term, (i) occurs once, as a lower index, and therefore follows the free-index occurrence pattern, while (j) occurs twice, once lower and once upper, and is consumed by summation.

Fixed Variance Across the Expression

The occurrence pattern also requires that a free index keep the same variance everywhere it appears in an equation. A free index that is lower on one side of an equation must be lower on the other side, and it must be lower in every term that is added to it.

Occurrence Pattern Across Multiple Terms

Matching Requirement in Sums

When several terms are added together to form a single tensor equation, the occurrence pattern requires that each term expose the identical collection of free indices, with identical names and identical variance, so that the sum transforms consistently as a single tensorial object.

C_{i} = A_{i} + B_{i}

Here the free index (i) recurs with the same lower variance in every term, satisfying the occurrence pattern required for the sum to be a legitimate tensor equation.

Violation by Mismatched Variance

If one term in a sum carried (i) as a lower index while another carried it as an upper index, the occurrence pattern would be violated, since the two terms would not respond identically to a change of basis, and the resulting sum would not represent a single well-defined tensor.

Occurrence Pattern in Multi-Index Expressions

Independent Tracking of Each Free Index

When an expression contains more than one free index, the occurrence pattern applies independently to each one. Each free index is tracked separately for occurrence count and variance, regardless of how many dummy indices are also present in the same expression.

T_{i k} = A_{i j} B^{j k}

Here (i) and (k) each occur once and follow the free-index occurrence pattern independently, while (j) occurs twice and is summed away, leaving a result whose two free indices, (i) and (k), determine its rank.

Diagnostic Use of the Pattern

The occurrence pattern serves as a direct diagnostic tool: counting the appearances of each index symbol within a term immediately reveals which indices are free and which are dummy, and checking that every free index recurs with matching name and variance across all terms confirms whether an expression is a legitimate tensor equation. This diagnostic use makes the occurrence pattern one of the most practical checks available when reading or constructing tensor notation.