10.9.4 Tensor Covector Component Index Update

Tensor Covector Component Index Update is the operation of relabeling and recomputing the free index of a covector's component array when a change of basis is applied, replacing the old index with a new one carrying a prime while simultaneously contracting the old component against the forward change-of-basis matrix to obtain the corresponding new numerical value. It describes the mechanical act of index bookkeeping that accompanies the covector component change rule, distinct from the arithmetic of the transformation itself, and it ensures that the notation for a covector's components always signals unambiguously which basis those components belong to.

What the Update Consists Of

Attaching a Prime to the Free Index

When a covector's components are transformed to a new basis, the free index labeling those components acquires a prime, marking every subsequent reference to that component as belonging to the new basis rather than the old one.

Recomputing the Value Through Contraction

Alongside the relabeling, the index update requires the actual numerical value of the new component to be recomputed by contracting the old components with the forward change-of-basis matrix, so that the prime attached to the index is matched by an actual change in the underlying value.

Necessity of Performing Both Parts Together

Relabeling Without Recomputation Is Misleading

Attaching a prime to an index without actually recomputing the corresponding value would produce a notation that falsely claims a component belongs to the new basis while still holding the old numerical value, creating an inconsistency between notation and content.

Recomputation Without Relabeling Is Ambiguous

Conversely, recomputing the value using the forward matrix without updating the index label would leave no notational trace that the component now refers to a different basis, making it impossible for a reader to distinguish the new value from the old one by inspection alone.

The Update as a Single Atomic Operation

Because of this mutual dependency, the index update is best understood as a single atomic operation combining both the relabeling and the recomputation, performed together every time a covector's components are re-expressed in a new basis.

Contrast With the Vector Index Update

Opposite Matrix Factor

The covector component index update uses the forward matrix for its recomputation, in direct contrast to the vector component index update, which uses the inverse matrix, a distinction that must be tracked carefully whenever both a vector and a covector appear in the same body of work.

Shared Notational Convention

Despite this difference in matrix factor, the notational convention of attaching a prime to indicate a change of basis is shared identically between the vector and covector index updates, so that the same visual cue applies regardless of which index type is being updated.

Extension to Repeated Changes

Sequential Index Updates

When a covector undergoes a sequence of basis changes, the index update is applied once at each stage, with the prime notation extended, for instance to a double prime, to indicate a component that has undergone two successive updates, each accompanied by its own contraction with the corresponding forward matrix.

Cumulative Effect Equivalent to a Single Update

The cumulative effect of several sequential index updates is equivalent to a single index update performed with the composed forward matrix relating the first and last bases directly, so that the intermediate steps need not be retained once the final component values are obtained.

Schematic Representation

The diagram shows the unprimed covector component transforming into the primed component, with the arrow representing the simultaneous relabeling of the index and recomputation of the value that together constitute the index update.