10.11 Tensor Higher Order Component Change Rule

Tensor Higher Order Component Change Rule is the extension of the tensor component transformation law to tensors carrying three or more indices, stating that the new components are obtained from the old components by contracting each individual index with its own appropriate matrix factor, an inverse matrix for every upper index and a forward matrix for every lower index, all applied simultaneously within a single expression involving as many matrix factors as the tensor has indices. It generalizes the rank-one rules for vectors and covectors and the rank-two rules for mixed, purely contravariant, and purely covariant tensors to arbitrarily many indices, without introducing any new principle beyond repeating the same index-by-index assignment.

Statement of the Rule

General Form for Arbitrary Rank

For a tensor with any number of upper indices and any number of lower indices, the higher order component change rule assigns one inverse matrix factor to each upper index and one forward matrix factor to each lower index, with every factor contracted against the corresponding index of the old components.

This example, for a tensor with two upper and two lower indices, illustrates the pattern that continues without modification regardless of how many additional indices are added.

Counting the Matrix Factors

The total number of matrix factors appearing in the higher order component change rule equals exactly the total number of indices carried by the tensor, with the split between inverse and forward factors matching the split between upper and lower indices.

Construction of the Rule From Simpler Cases

Building Up From Rank-One Rules

The higher order rule can be viewed as the simultaneous, independent application of the vector component change rule to each upper index and the covector component change rule to each lower index, combined into a single tensor equation rather than treated as separate equations for separate objects.

Consistency With the Rank-Two Cases

Restricting the higher order rule to exactly two indices reproduces precisely the mixed, purely contravariant, or purely covariant rank-two rules, depending on which combination of upper and lower indices is present, confirming that the rank-two cases are themselves special instances of the same general pattern.

Properties Preserved by the Rule

Tensor Preservation at Any Rank

Just as with rank-one and rank-two tensors, applying the higher order component change rule together with the corresponding transformation of the basis vectors and dual basis covectors leaves the reconstructed tensor object completely unchanged, regardless of how many indices it carries.

Independent Contraction of Each Index

Because each index contributes its own matrix factor through its own independent contraction, operations that act on only some of the indices of a higher-rank tensor, such as a partial contraction, transform consistently with the corresponding lower-rank rule for the remaining free indices.

Composition Under Successive Basis Changes

Applying the higher order component change rule for a sequence of basis changes composes in exactly the same way as for lower-rank tensors, with the matrix factors for each individual change of basis multiplying together according to the order in which the changes are applied.

Schematic Representation

The diagram represents the higher order component change rule as a single arrow connecting old and new component arrays, with as many matrix factors involved as the tensor has indices, applied all at once.