10.21.2 Tensor Change Verification Target Basis

Tensor Change Verification Target Basis is the specific step, within the broader change-of-basis verification procedure, of confirming that the target chart's basis vectors, produced as the output of the transformation, are themselves linearly independent and properly paired with their own dual basis, ensuring that the newly computed components are being expressed relative to a genuinely valid basis rather than one that has degenerated as an unintended consequence of the transformation.

What Must Be Confirmed About the Target Basis

Linear Independence of the Resulting Basis Vectors

Just as the source basis was required to be linearly independent, the target basis vectors $${\overline{e}}_j$$, obtained from the source basis through the forward Jacobian, must also be checked for linear independence at the point of interest:

Equivalence to Jacobian Non-Singularity

Because the target basis is obtained from the source basis via the forward Jacobian, this independence check is mathematically equivalent to confirming that the forward Jacobian itself is non-singular at the point being examined, so this step can be carried out directly by computing the Jacobian determinant rather than working with the basis vectors as abstract objects.

Confirming the Target Dual Basis

Correct Pairing With the New Primary Basis

If covariant components have been computed in the target chart, the verification confirms that the corresponding target dual basis, obtained from the source dual basis via the inverse Jacobian, still satisfies the defining pairing relation with the new primary target basis:

Guaranteed by Correct Application of the Rule

If the earlier verification steps in the procedure, particularly the matrix inverse check, have already succeeded, this pairing relation is guaranteed to hold automatically as a consequence of the Jacobian product identity, so this step largely serves as a final direct confirmation that no error was introduced specifically in constructing the target basis and target dual basis themselves.

Diagram of Target Basis Verification

Confirming Validity of the Output Basis

Why Target Basis Degeneration Can Occur

Coordinate Singularities Introduced by the Map

Even when the source basis is perfectly valid, a poorly chosen or singular target chart, such as a curvilinear coordinate system with a coordinate singularity at a specific point, can produce a degenerate target basis at exactly that point, making the target basis verification step necessary even when the source basis has already been separately confirmed as valid.

Errors Introduced During the Transformation Arithmetic

An arithmetic error made while computing the forward Jacobian entries themselves, distinct from an error in the inverse, can produce a target basis that appears independent under casual inspection but fails a rigorous determinant check, which is why this step relies on an explicit numerical or symbolic test rather than a visual or intuitive judgment of the resulting basis vectors.

Placement Within the Broader Procedure

Confirming the Output, Not Just the Input

While the source basis verification step guards against beginning the transformation with already-flawed data, the target basis verification step serves the complementary role of confirming that the transformation has not itself produced degenerate or invalid output, closing the loop on basis validity at both ends of the change-of-basis computation.

Enabling Confident Reuse of the Target Data

Once the target basis has been confirmed valid, the newly computed tensor components in the target chart can be used with the same confidence as the original source components in any further computation, including as the source data for a subsequent change of basis, without needing to re-derive their validity from scratch.