3.4.2 Tensor Dual Basis Index Matching

Tensor Dual Basis Index Matching is the convention, arising directly from the defining relation e^i(e_j) = δ^i_j, that an upper index and a lower index sharing the same letter are understood to be tested against one another through the Kronecker delta, and the systematic use of this convention, together with the Einstein summation rule, to read and manipulate tensor expressions without repeatedly writing out explicit basis evaluations or summation symbols.

The Kronecker Delta as an Index-Matching Device

What δ^i_j Encodes

The Kronecker delta δ^i_j, equal to 1 when i = j and 0 otherwise, is precisely the array of values obtained by testing each dual basis vector e^i against each basis vector e_j. Writing e^i(e_j) = δ^i_j packages the entire dual basis relation into a single symbol whose behavior is fully described by whether its upper and lower indices coincide.

Matching as Selection

Because δ^i_j vanishes unless i = j, any sum Σ_j X_j δ^i_j collapses to the single term X_i, since every other term in the sum is multiplied by zero; the Kronecker delta acts as a selector, picking out exactly the term whose lower index matches the upper index i and discarding all others. This selection behavior is the mechanical content underlying every computation that uses the dual basis relation.

Index Matching in the Einstein Summation Convention

Repeated Upper-Lower Pairs Imply Summation

Under the Einstein summation convention, an index appearing once as an upper index and once as a lower index within a single term is automatically summed over, without an explicit Σ symbol: v^i e_i means Σ_i v^i e_i, and ω_i v^i means Σ_i ω_i v^i. This convention is a direct extension of dual basis index matching, since the natural pairing ω(v) = Σ_i ω_i v^i is exactly a sum over an index that appears once upper (on v^i) and once lower (on ω_i).

Why Only Upper-Lower Pairs Are Summed

The convention specifically restricts automatic summation to a repeated index appearing once upper and once lower, rather than any repeated index regardless of position, because this is exactly the pattern guaranteed to produce a basis-independent result: the coordinate transfer rules for upper and lower indices are inverse to one another, so an upper-lower contracted pair is invariant under change of basis, while two repeated upper indices or two repeated lower indices would not, in general, produce an invariant quantity.

Index Matching Across Multiple Tensor Factors

Matching in Contraction

For a (p, q)-tensor T and a chosen upper index position and lower index position, contraction sums over a shared index letter placed in both positions, C(T)^{...}_{...} = T^{...i...}_{...i...}, again relying on index matching to specify exactly which upper slot is being paired against which lower slot; the choice of which two slots to contract is communicated entirely by which letter is repeated between an upper and a lower position.

Matching in the Coordinate Transfer Rule

The general coordinate transfer formula for a (p, q)-tensor, described in tensor isomorphism coordinate transfer, involves summing over matched pairs of old and new indices, each transition-matrix factor a or b carrying one upper and one lower index that must align correctly with the corresponding index of the tensor being transformed; correct index matching in this formula is what guarantees that contracted quantities, such as full tensor contractions, come out invariant under the change of basis, as shown by the cancellation Σ_k b^i_k a^k_j = δ^i_j.

Practical Discipline of Index Matching

Free Indices Versus Summed Indices

In any tensor expression, an index appearing exactly once, with no matching partner of opposite variance, is a free index, labeling one component of the resulting tensor, while an index appearing exactly twice, once upper and once lower, is a summed, or dummy, index, and disappears from the final expression; correctly distinguishing free indices from summed indices, based purely on this upper-lower matching pattern, is the basic literacy skill required to read and write tensor equations in index notation.

Consistency Requirement

A well-formed tensor equation must have exactly the same set of free indices, in the same upper or lower positions, on both sides of an equality; violating this requirement, for instance equating a quantity with a free upper index i to a quantity with a free lower index i, signals an expression that cannot be correct, since the two sides would transform differently under a change of basis and could not represent the same invariant relationship.