9.14.5 Tensor Basis Change Tensor Preservation

Tensor Basis Change Tensor Preservation maintains tensor invariance under basis transformations, ensuring mathematical consistency across coordinate systems.

Tensor Basis Change Tensor Preservation is the property guaranteeing that a tensor, as an abstract object, remains exactly the same before and after its components are transformed from one basis to another. It is the condition that justifies calling a basis change a change of representation rather than a change of the tensor itself.

Statement of Preservation

The Tensor Does Not Depend on the Basis

A tensor is defined independently of any particular basis; the basis is only a tool used to express the tensor through numerical components. Preservation asserts that recovering the tensor from its transformed components in the new basis gives back the identical tensor recovered from the original components in the old basis.

\sum_{i, j} T_{j}^{i} e_{i} \otimes e^{j} = \sum_{i, j} {\bar{T}}_{j}^{i} {\bar{e}}_{i} \otimes {\bar{e}}^{j}

Preservation as the Purpose of the Transformation Rules

The specific rules by which contravariant and covariant components transform, using the transformation matrix and its inverse respectively, are derived precisely so that this equality holds. Preservation is not a coincidental byproduct; it is the requirement that determines what the transformation rules must be.

Why Preservation Holds

Cancellation Between Basis and Components

Preservation follows from the fact that basis vectors transform using the transformation matrix while contravariant components transform using its inverse, so that the two transformations cancel exactly when combined in the summation form. Covariant components and dual basis covectors cancel in the analogous, opposite way.

Consistency of the Duality Relation

The transformation of the dual basis is constructed specifically to maintain the duality relation with the new basis vectors. This constructed consistency is what allows covariant components to transform correctly and contributes to the overall preservation of the tensor.

Consequences of Preservation

Invariant Scalars

Any full contraction of a tensor's indices, producing a plain scalar, yields the same numerical value regardless of which basis was used to compute the components entering the contraction. This invariance of scalars is a direct consequence of tensor preservation under basis change.

Physical and Geometric Meaning

In applications where tensors represent physical or geometric quantities, preservation is what allows such a quantity to have an objective meaning independent of the observer's or analyst's choice of coordinate axes, even though the numerical components used to describe it differ from one choice of axes to another.

Verifying Preservation

Direct Substitution Check

Preservation can be verified directly by substituting the transformation rules for components and basis vectors into the summation form and confirming, through the algebra of the transformation matrix and its inverse, that all transformation factors cancel to leave the original expansion unchanged.

Failure Indicates an Incorrect Transformation

If applying a proposed transformation to a tensor's components does not preserve the tensor when recovered in the new basis, this indicates that the transformation rule used was not the correct one for the tensor's type, most commonly a mismatch between using the transformation matrix and its inverse for contravariant versus covariant indices.