10.7.4 Tensor Component Law Summation Pattern

The Tensor Component Law Summation Pattern describes how tensor components change under coordinate transformations through indexed summation rules.

Tensor Component Law Summation Pattern is the specific arrangement of repeated indices within the tensor component transformation law that instructs the Einstein summation convention over which range of values the implicit sum must be carried out, occurring once for every index the tensor carries and involving exactly one contraction between a matrix factor and the corresponding original component. It is the recurring structural feature that turns the abstract statement of the transformation law into an explicit, computable sum once a dimension for the underlying vector space has been fixed.

Anatomy of the Pattern

One Contraction Per Index

For every index of the tensor being transformed, the summation pattern introduces one repeated letter, appearing as an upper index on one factor and a lower index on another, signaling that a sum must be taken over all values that index can assume, from one up to the dimension of the vector space.

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

Here the letter serving as the summation index appears as a lower index on the inverse matrix factor and as an upper index on the original component, marking the single contraction associated with the one index this vector carries.

Independence of Separate Summation Letters

When a tensor carries more than one index, the summation pattern uses a distinct letter for each separate contraction, ensuring that the sums associated with different indices remain independent of one another and do not interfere.

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

In this two-index example, one summation letter links the inverse matrix factor to the first index of the original tensor, while a separate summation letter links the forward matrix factor to the second index, with the two sums carried out independently and then multiplied together term by term.

Governing Rules

Range of Summation Set by Dimension

The summation pattern implies that each repeated index ranges over every integer from one up to the dimension of the underlying vector space, so that the explicit sum expands into as many terms as that dimension, all added together to yield a single component value.

Exactly One Repetition Per Summed Letter

A valid summation pattern never repeats the same letter more than twice within a single term, since a third occurrence would make it impossible to tell which pair of factors is meant to be contracted. The pattern therefore always pairs each summed letter with precisely one upper occurrence and one lower occurrence.

No Summation Over Free Indices

Indices that appear only once within a term, without a matching occurrence elsewhere, remain free and are excluded from the summation pattern entirely. These free indices instead label which particular component of the transformed tensor the equation describes.

Consequences of the Pattern

Expansion Into an Explicit Sum

Once a dimension is fixed, the summation pattern can be expanded directly into a finite sum with as many terms as the dimension, replacing the implicit repeated-index notation with an explicit addition of individually computed products.

Compact Representation of Large Computations

Because the summation pattern condenses what would otherwise be a lengthy explicit sum into a single repeated index, it allows the component transformation law to be written compactly regardless of how large the dimension of the vector space happens to be, without changing the underlying computation being described.

Schematic Representation

The diagram highlights the shared summation letter linking a matrix factor to a component, representing the implicit sum that the summation pattern instructs the reader to carry out over that repeated index.