11.20.2 Tensor Variance Verification Transformation Factor

The Tensor Variance Verification Transformation Factor ensures tensor variance consistency under coordinate changes.

Tensor Variance Verification Transformation Factor is the specific stage of the verification procedure that isolates the exact matrix or Jacobian factor produced when a candidate object is transformed under an explicit basis or coordinate change, comparing that isolated factor directly against the precise factor required by the claimed variance type rather than accepting a superficial resemblance between the two.

Foundational Setting

Isolating the Factor as a Distinct Object

Where the index position check asks whether an index is placed correctly, the transformation factor check goes further, extracting the actual matrix expression that multiplies the original components after a basis change and treating that expression as an object to be compared symbolically against the theoretically required factor.

The Two Candidate Factors

For any tensor claim, there are exactly two admissible transformation factors under a basis change described by matrix $A$ : the direct matrix $A$ itself, associated with covariant behavior, and its inverse $A^{- 1}$ , associated with contravariant behavior. The verification check determines which of these two factors, if either, genuinely appears.

Procedure for the Factor Check

Step One: Derive the Raw Transformation

Starting from the definition of the candidate quantity, compute its new-basis components explicitly in terms of the old-basis components and the entries of $A$ , without presupposing which factor will emerge.

Step Two: Algebraically Isolate the Factor

Rearrange the resulting expression into the form of a single matrix, or matrix inverse, multiplying the old components, isolating this coefficient as an explicit factor:

{\tilde{v}}^{i} = \sum_{j} (isolated factor) v^{j}

Step Three: Compare the Isolated Factor to the Required Factor

Confirm whether the isolated factor equals $A$ , equals $A^{- 1}$ , or matches neither, in which case the object fails to qualify as a tensor of either simple variance type.

Worked Comparison

Contravariant Confirmation

For an ordinary displacement vector under a linear basis change, the isolated factor is found to satisfy:

isolated factor = A^{- 1}

confirming contravariant behavior and validating the upper-index notation.

A Failing Case

For connection coefficients, the isolated factor does not reduce to either $A$ or $A^{- 1}$ alone but includes an additional inhomogeneous term, so the transformation factor check correctly reports that these coefficients are not a tensor.

Visual Overview

Diagram of the Factor Comparison

Extending the Check to Coordinate Transformations

Isolating the Jacobian Factor

In the coordinate setting, the analogous check isolates the Jacobian-based factor arising from an explicit coordinate change and compares it against the Jacobian $J$ built from new coordinates with respect to old, or its counterpart $K$ built in the reverse direction, exactly mirroring the abstract basis case.

Consistency Across Multiple Coordinate Patches

When a quantity is defined across several overlapping coordinate patches, the transformation factor check can be repeated for each pair of overlapping patches, confirming that the same admissible Jacobian factor, and not some patch-dependent variant, governs the transition everywhere the patches overlap.

Summary of Key Traits

Defining Characteristics

The transformation factor check isolates the explicit matrix or Jacobian coefficient produced by a basis or coordinate change.
This isolated factor is compared directly against the two admissible tensor factors, the direct matrix and its inverse.
A match with one factor confirms the corresponding variance type; a match with neither indicates the object is not a simple tensor.
The check extends naturally from abstract basis changes to Jacobian-based coordinate transformations, including across overlapping coordinate patches.