8.22.1 Tensor Indexed Equation Free Index Set

Tensor Indexed Equation Free Index Set is the collection of all index letters that appear exactly once, with a definite variance, on each side of a tensor equation written in index notation — the letters that are not contracted away by the Einstein summation convention and that therefore remain to label the components of the equation's left- and right-hand sides. It is the primary structural feature checked when validating a tensor equation, since a well-formed equation must have identical free index sets, with matching variance, on both sides, before any question of whether the equation is numerically true can even be posed.

Identifying the Free Index Set

Scanning for Single Occurrences

Given an indexed equation, the free index set of one side is found by listing every index letter appearing on that side and discarding any letter that occurs twice (once upper, once lower) within a single term, since such a letter is a dummy index consumed by contraction. The letters that remain, each occurring exactly once, constitute the free index set:

has free index set {i} on both sides; j is a dummy index on the right-hand side and does not belong to the free index set at all.

Free Index Sets Are Computed Per Side

Because the same letter could in principle be used differently in different terms of a larger sum of terms, the free index set is properly determined term by term and then compared across all terms that are added together, requiring that every additive term making up one side of the equation share the identical free index set before that side's overall free index set is well defined.

The Matching Requirement Across an Equation

Both Sides Must Share the Same Free Index Set

A tensor equation is only well-formed if the free index set of its left-hand side is identical, letter for letter and variance for variance, to the free index set of its right-hand side. An expression such as Tⁱ = A^i_j B^j + Cᵏ is not a valid tensor equation as written, because the free index set of the right-hand side, {i, k}, does not match the free index set {i} of the left-hand side — the stray free index k has no counterpart to equate it to.

is the basic balance condition every legitimate tensor equation must satisfy.

Variance Must Match, Not Just the Letter

It is not enough for the same letter to appear on both sides; it must appear with the same variance. An equation with i upper on the left and i lower on the right is not properly balanced as a tensor equation, since the two sides would then transform differently under a change of basis even though they share a common index letter — the free index set comparison is therefore a comparison of upper/lower-labeled letters, not of bare letters.

What the Free Index Set Represents

One Component Equation per Element Combination

The free index set of a balanced equation specifies exactly how many separate scalar equations the compact indexed equation stands for: one scalar equation for every combination of values that the indices in the free index set can independently take. An equation with free index set {i} in n dimensions represents n scalar equations, while an equation with free index set {i, k} represents n² scalar equations, one for each pair of values.

The Free Index Set Determines the Result's Type

The size and variance composition of the free index set fixes the tensor type of whatever the equation is computing: an equation whose both sides have free index set consisting of two upper letters and one lower letter describes a (2,1) tensor equation, and any candidate right-hand side proposed for such an equation must itself reduce, after all contractions are accounted for, to exactly that same free index set.

Diagram of a Balanced Versus Unbalanced Free Index Set

Practical Use of the Free Index Set Check

First Diagnostic Applied to a Candidate Equation

Because checking the free index set requires no knowledge of the actual values or meaning of a tensor equation, it is the first and cheapest diagnostic applied when a proposed tensor identity is written down: any mismatch in the free index sets of the two sides immediately signals a notational error — a missing contraction, a mistyped index, or an inconsistent choice of variance — well before any deeper investigation of whether the equation could be true is warranted.

Guiding the Construction of New Identities

When deriving a new tensor identity from known ones, tracking the free index set at each step ensures that intermediate manipulations remain well-formed: any operation applied to one side of an equation, such as multiplying both sides by an additional tensor factor or contracting an additional index, must be applied in a way that keeps the free index sets of both sides equal after the operation, which is the discipline that prevents an otherwise plausible-looking derivation from silently becoming ill-formed partway through.