11.9.4 Tensor Mixed Law Multi Index Update

Tensor Mixed Law Multi Index Update is the complete procedure by which every index of a mixed tensor, whether upper or lower and regardless of how many of each are present, is simultaneously recalculated in a new coordinate system by contracting each old index with its appropriate direct or inverse Jacobian factor, producing the fully transformed component with the new coordinate labels attached.

Definition and General Procedure

Assembling the Full Update From Individual Factors

The multi index update is constructed by assigning one direct Jacobian factor to each upper index and one inverse Jacobian factor to each lower index of the tensor, then multiplying all of these factors together and summing over every old index that appears, producing a single new component for each choice of new index values.

Distinct Summation Variables for Each Index

Each old index appearing in the update is given its own summation variable, distinct from the summation variables used for the other indices, so that the update sums over each index independently and no cross-terms between different index positions are introduced.

Structural Properties of the Full Update

Simultaneous Rather Than Sequential in Effect

Although the update can be conceptually broken into separate upper index updates and lower index updates applied one at a time, the multi index update produces the identical final result whether the individual index updates are carried out in sequence or all performed together as one combined contraction, since the operations act on independent indices.

Scaling With the Number of Indices

The number of Jacobian factors appearing in the multi index update equals the total number of indices on the tensor, upper and lower combined, so a tensor with more indices requires proportionally more factors in its transformation, though each individual factor remains either the direct or the inverse Jacobian matrix depending on the index type.

Practical and Consistency Considerations

Correctness Verified Through the Identity Case

When the new coordinate system is identical to the old one, every direct Jacobian factor and every inverse Jacobian factor reduces to the appropriate Kronecker delta, and the multi index update leaves all components exactly unchanged, providing a basic check that the procedure has been assembled correctly.

Compatibility With Partial Updates

Because the multi index update decomposes cleanly into independent per-index contractions, it is possible to update only a subset of indices at an intermediate stage of a calculation, provided that the final result still applies every remaining factor before the tensor is considered fully transformed to the new coordinate system.

Role Within Tensor Algebras

General Statement of the Mixed Variance Law

The multi index update is simply the mixed variance transformation law stated in its most general form, applicable to a tensor of any type, and every more specific case, including the pure covariant law, the pure contravariant law, and mixed tensors with only one index of each kind, is obtained from it by setting the appropriate index counts.

Foundation for Computational Implementation

Because the multi index update is built from a well-defined product of independently computable factors, it provides a systematic basis for computing the transformation of tensors of arbitrary rank in symbolic or numerical calculations, without requiring a separate derivation for each new combination of upper and lower indices.