10.7.3 Tensor Component Law Index Placement

Tensor Component Law Index Placement explains how tensor components transform under coordinate changes through index placement, showing vector and covector behavior.

Tensor Component Law Index Placement is the arrangement of upper and lower indices on a tensor symbol and on its associated change-of-basis matrix factors within the component transformation law, fixed so that every summed index appears exactly once as an upper index and once as a lower index, and every free index of the transformed component matches, in position and letter, the corresponding free index on the original component. It is the structural discipline that makes the component transformation law readable and mechanically checkable at a glance, since a correctly placed index pattern signals immediately which contractions are intended and which indices remain free.

Rules Governing Placement

Free Indices Must Match Across the Equation

Every free index appearing on the transformed component on one side of the transformation law must appear, with the same letter and the same upper or lower position, on the transformed component on the other side, ensuring that the equation compares like with like.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Here the free index on the left, attached to the transformed component, matches in letter and position the upper index carried by the matrix factor on the right, while the summed index appears once up and once down.

Summed Indices Must Appear Once Up and Once Down

Any index that is contracted, meaning it is repeated within a single term, must occur exactly once as an upper index and exactly once as a lower index somewhere within that term. An index repeated twice in the same position, both upper or both lower, signals a violation of the placement rule and typically indicates an error in the formula.

Distinct Letters for Distinct Indices

Within a single application of the component transformation law, every index that is not meant to be summed with another must be given a distinct letter, so that no ambiguity arises about which indices are free and which are being contracted.

Placement for Different Index Types

Contravariant Index Placement

An upper, contravariant index on the original component is placed as an upper index on the transformed component as well, and it is linked through a shared summation letter to the lower index of the inverse matrix factor responsible for transforming it.

Covariant Index Placement

A lower, covariant index on the original component is placed as a lower index on the transformed component, linked through a shared summation letter to the upper index of the forward matrix factor responsible for transforming it.

ω_{i}' = A_{i}^{j} ω_{j}

Mixed Index Placement for Higher-Rank Tensors

For a tensor carrying both upper and lower indices simultaneously, the placement rule is applied separately and independently to each index, with each summation letter used only once across the entire expression to avoid conflating distinct contractions.

T_{j}^{i}' = {(A^{- 1})}_{k}^{i} A_{j}^{l} T_{l}^{k}

Consequences of Correct Placement

Immediate Verification of Well-Formedness

A correctly placed index pattern allows an expression to be checked for internal consistency purely by inspection, without evaluating any numerical values, since any repeated index not appearing once up and once down, or any mismatch of free indices across an equation, immediately signals a malformed expression.

Enabling the Summation Convention

Consistent index placement is what allows the summation convention to be applied unambiguously, since the convention relies entirely on recognizing repeated upper and lower index letters as an instruction to sum, a recognition that fails if the placement rules are not followed.

Schematic Representation

The diagram highlights the free index carried across the equation and the summed index appearing once as an upper index and once as a lower index, the two patterns that together define correct index placement within the component transformation law.