10.11.2 Tensor Higher Order Component Indexwise Update

Tensor Higher Order Component Indexwise Update is the procedure of relabeling and recomputing every index of a higher-rank tensor's component array, one index at a time, when a change of basis is applied, so that each index individually receives a prime and is contracted against its own appropriate matrix factor, with the collection of all these individual updates performed together producing the fully transformed component. It generalizes the single-index update used for vectors and covectors, and the two-index update used for mixed rank-two tensors, to tensors of any rank, treating the update of a many-indexed object as the coordinated combination of many simple, individually understood single-index updates.

Structure of the Indexwise Update

Updating Each Index According to Its Own Type

Every index of the tensor is examined individually, and depending on whether it is upper or lower, it is relabeled with a prime and contracted with the inverse matrix or the forward matrix respectively, exactly as it would be if that index belonged to a vector or a covector on its own.

Here each of the four indices, two upper and two lower, undergoes its own individual contraction, with the four resulting factors combined together into one single expression for the fully updated component.

Simultaneity of the Individual Updates

Although each index is updated according to its own rule, all of the individual updates are carried out together within a single equation rather than in a sequence of separate steps, since the final component depends on all indices simultaneously through the shared summation over each contracted pair.

Necessity of Consistent Indexwise Treatment

Partial Updates Produce Inconsistent Objects

Updating some indices of a higher-rank tensor while leaving others unprimed would produce a hybrid expression that refers to the new basis in some of its indices and the old basis in others, an object that no longer corresponds to any single, well-defined component in either basis.

Correct Matrix Assignment Per Index Is Mandatory

Because each index carries its own type, applying the wrong matrix factor to even a single index, for instance using the forward matrix on an upper index that should receive the inverse matrix, breaks the tensor preservation property for the entire component, even if every other index was updated correctly.

Relationship to Lower-Rank Updates

Reduction to Simpler Cases

Restricting the indexwise update to a tensor with only one or two indices reproduces exactly the single-index updates used for vectors and covectors, or the two-index update used for mixed rank-two tensors, confirming that those simpler updates are themselves special cases of the general indexwise procedure.

Extension Without New Principles

No new principle is introduced when extending the indexwise update to tensors with more than two indices; the same per-index assignment of matrix factors, already established for lower-rank tensors, is simply repeated once for every additional index the tensor carries.

Schematic Representation

The diagram shows four indices of a higher-rank tensor, each independently assigned its own matrix factor according to type, illustrating the coordinated but individually determined nature of the indexwise update.