7.15.3 Tensor Component Index Transformation Rule

The Tensor Component Index Transformation Rule explains how tensor components change with coordinate systems, key for physics and math applications.

Tensor Component Index Transformation Rule is the precise mathematical formula specifying, for a single index of a tensor, exactly which factor drawn from the transition matrix or its inverse must be applied, and how it must be summed against the old components, in order to obtain that index's contribution to the new components under a change of basis.

The Rule for a Single Contravariant Index

Applying the Inverse Transition Matrix

A contravariant index transforms according to a rule that uses the inverse of the transition matrix, summing over the old index it replaces, so that the new component along a given direction is built from a weighted combination of all the old components.

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

Reading the Rule's Index Pattern

In this rule, the new primed index labels which row of the inverse transition matrix is used, while the old unprimed index runs over every column of that same matrix and simultaneously over every old component, with the sum collecting the full contribution across all old directions.

The Rule for a Single Covariant Index

Applying the Transition Matrix Directly

A covariant index instead transforms using the transition matrix itself, rather than its inverse, again summing over the old index it replaces, reflecting the opposite direction of response compared to a contravariant index.

ω_{i^{'}} = \sum_{j = 1}^{n} A_{i^{'}}^{j} ω_{j}

Symmetry Between the Two Rules

The contravariant and covariant index transformation rules share the identical structural pattern, a single sum weighted by one matrix factor, differing only in whether that factor is drawn from the transition matrix or from its inverse, a symmetry that makes the two rules easy to remember together.

Applying the Rule Index by Index

One Rule Application Per Index

For a tensor carrying several indices, the full transformation is obtained by applying the appropriate single-index rule once for each index the tensor carries, using the contravariant version for every upper index and the covariant version for every lower index.

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

Independence of the Rule From Other Indices

The transformation rule applied to any one index does not depend on how many other indices the tensor carries or on their variance type, meaning the same single-index rule extends unchanged from the simplest vector or covector case up through tensors of arbitrarily high rank.

Verifying an Application of the Rule

Checking Index Placement

A correct application of the rule places the new primed index as the target of the matrix factor and sums the old unprimed index against the corresponding old component, and reversing this placement by mistake produces an incorrect, mismatched transformation.

Confirming Recoverability of the Original

Applying the transformation rule with the transition matrix and then applying it again with the inverse relationship should recover the original components exactly, providing a direct check that the rule has been applied correctly in a given calculation.

Diagrammatic Illustration

The transformation rule pictured as a weighted rerouting of each old component's contribution into the corresponding new component.

Broader Role of the Index Transformation Rule

Building Block for the Full Transformation Law

The single-index transformation rule is the fundamental unit from which the complete transformation law for any tensor, regardless of rank, is assembled, making a solid grasp of this one rule sufficient to construct the transformation for tensors of arbitrary complexity.

Guarantee of Coordinate-Independent Meaning

Correct and consistent application of the index transformation rule across every index of a tensor is precisely what guarantees that quantities built from the tensor, such as full contractions, retain a meaning that does not depend on the arbitrary choice of basis used to describe them.