8.22.3 Tensor Indexed Equation Term Structure

Tensor Indexed Equation Term Structure is the internal organization of a single additive term within a tensor equation into its constituent tensor factors, each factor's own indices, and the pattern of index sharing across factors that determines which indices are free and which are contracted within that term. It describes what a term is built from — a product of one or more tensor components, each carrying its own indices, combined according to the repeated-index summation rule — as distinct from the equation-wide question of how multiple such terms are added together and matched across an equals sign.

Anatomy of a Single Term

Factors and Their Individual Indices

A term such as g_{ij} A^i_{\ k} B^{jk} is built from three tensor factors, g_{ij}, A^i_{\ k}, and B^{jk}, each contributing its own indices with its own variance. The term's overall structure is the specification of exactly which factors are multiplied together and which index letters, among all the indices carried by these factors, coincide with one another.

Here i, j, and k each occur exactly twice across the whole term, once upper and once lower in every case, so all three are dummy indices contracted away within this single term, leaving no free index for this particular term as written.

Index Sharing Determines the Contraction Pattern of the Term

The term structure is precisely the pattern of which index letters are shared between which factors: i links g_{ij} to A^i_{\ k}, j links g_{ij} to B^{jk}, and k links A^i_{\ k} to B^{jk}, so the term describes a three-way chain of contractions rather than a simple pairwise contraction, and this chain structure is exactly what the term's index pattern records.

Coefficient and Scalar Multiples Within a Term

Numerical or Symbolic Coefficients Attach to the Whole Term

A term may carry a scalar coefficient, such as 2 A^i_j B^j or (1/2) g^{ij} h_{ij}, and this coefficient is understood to multiply the entire product of tensor factors rather than any single factor's individual components; the coefficient does not participate in the index structure and is unaffected by which indices are free or contracted within the term.

Sign as a Structural Element

The sign attached to a term — whether it is added or subtracted in the larger equation — is likewise a feature of the term as a whole and is tracked separately from its index pattern; two terms with identical tensor factors and identical index-sharing pattern but opposite signs, such as A^i_j B^j and −A^i_j B^j, have the same term structure in the index sense even though they contribute oppositely to the sum.

Comparing Term Structures Across an Equation

Every Term Contributing to One Side Must Share the Same Free Indices

When several terms are added together to form one side of an indexed equation, the term structure of each individual term must, considered on its own, produce the same free index set as every other term in that sum, even though the terms may differ arbitrarily in which tensors they involve or how those tensors are internally contracted. A sum such as A^i_j B^j + C^{ik}D_k is well-formed at the level of term structure because both terms independently reduce to the free index i alone, despite using entirely different tensor factors and different dummy-index letters internally.

Term Structure Mismatches Signal Errors

If one term in a sum reduces, by its own internal index pattern, to a different free index set than the other terms it is added to, the overall sum is not a well-formed tensor expression regardless of how correct each individual term's internal contraction pattern might be; this term-by-term check of free index sets is precisely how the broader equation-level free index set requirement is verified in practice, one additive term at a time.

Diagram of a Term's Internal Contraction Chain

Distinguishing Term Structure From Equation-Level Structure

A Local Versus a Global Property

Term structure concerns only what happens inside one product of tensor factors — its own contraction chain, its own coefficient and sign; the free index set of an equation, by contrast, is a global property comparing the combined result of every term on one side against every term on the other. Analyzing an equation correctly requires both: verifying that each term's internal structure is self-consistent, and then verifying that the free indices surviving from every term line up with one another across the whole equation.

Term Structure as the Unit of Substitution

When substituting one tensor expression in place of another within a larger equation — for instance, replacing an intermediate result with its own defining formula — the substitution is performed one term at a time, respecting each term's internal structure, which is why understanding term structure in isolation, apart from the full equation, is a necessary step in any nontrivial tensor derivation carried out in index notation.