10.7.5 Tensor Component Law Tensor Preservation

Tensor Component Law Tensor Preservation is the guarantee that applying the tensor component transformation law to the components of a tensor, together with the corresponding change to the basis vectors and dual basis covectors, leaves the reconstructed tensor object exactly unchanged, so that the abstract multilinear object being described remains the same regardless of which basis its components happen to be written in. It is the property that justifies calling the transformation law a law of a tensor rather than an arbitrary rule for relabeling numbers, since it certifies that the numbers being transformed still refer to the identical underlying object after the transformation as before it.

Statement of Preservation

Reconstruction From Components and Basis

A tensor is reconstructed from its components by pairing each component with the corresponding combination of basis vectors and dual basis covectors. Tensor preservation asserts that this reconstruction yields the same result whether the old components are paired with the old basis or the new components, produced by the component transformation law, are paired with the new basis.

Both expressions on the right describe the identical object on the left, illustrating that the component transformation law, applied together with the basis change, preserves the vector being represented.

Extension to Tensors of Arbitrary Rank

The same equality holds for tensors of any rank, with each term in the reconstruction involving as many basis vectors or dual basis covectors as the tensor has indices, and the component transformation law is precisely what makes this equality hold for every index simultaneously.

Why Preservation Holds

Cancellation Between Matrix Factors and Basis Change

Preservation follows because the matrix factor applied to a component and the matrix factor implicit in the corresponding change of basis vector are inverses of one another, so that when the transformed component is paired with the transformed basis, the two matrix factors cancel and only the original pairing survives.

Carrying out the sum over the shared index reduces the product of the inverse matrix and the forward matrix to the identity, leaving the original component paired with the original basis vector.

Necessity of Using Matched Inverse Factors

Preservation depends critically on the component transformation law assigning the inverse matrix to contravariant indices precisely because the basis vectors themselves transform with the forward matrix. Any deviation from this matched assignment would break the cancellation and destroy the preservation property.

Consequences of Preservation

Justification for Calling the Law a Tensor Law

Because preservation holds, the component transformation law can be applied to any candidate set of indexed numbers as a test: if the numbers transform according to the law and the reconstructed object remains unchanged, they qualify as genuine tensor components rather than an arbitrary array that merely resembles one in a single basis.

Basis Independence of Physical and Geometric Statements

Tensor preservation is what allows equations written entirely in terms of tensors to hold in every basis simultaneously, since each individual tensor appearing in such an equation is guaranteed to represent the same object regardless of the basis chosen to express its components.

Foundation for Chained Transformations

Because preservation holds for any single valid change of basis, it also holds when several changes of basis are composed in sequence, since each individual step preserves the tensor and the composition of preserving steps must itself preserve the tensor.

Schematic Representation

The diagram shows two distinct reconstructions, one using old components with the old basis and one using new components with the new basis, converging on the identical tensor, the essence of tensor preservation under the component transformation law.