10.17.1 Tensor Active Transformation Object Change

Tensor Active Transformation Object Change describes how tensors change under active transformations, altering their components while preserving their geometric meaning.

Tensor Active Transformation Object Change is the genuine alteration of a tensor as a geometric or physical object, produced when an active transformation is applied while the coordinate basis is held fixed, resulting in a second, generally distinct tensor whose properties, such as direction, magnitude, or symmetry, may differ from those of the original.

What Changes Under an Active Transformation

The Tensor Itself Is Replaced

Unlike the passive case, where a single tensor is merely re-described in a new basis, an active transformation produces a new tensor:

V^{' i} = \sum_{k}^{n} A_{k}^{i} V^{k}

with $V^{' i}$ the components, in the single fixed basis used throughout, of a tensor that is, in general, a different object from the one with components $V^{i}$ , not merely the same object described differently.

Verifying Genuine Change

A direct way to confirm the object has genuinely changed, rather than merely being re-described, is to check that the fixed basis is used consistently for both the original and the transformed components; since no basis change has occurred, any numerical difference between the two component sets reflects an actual difference between two distinct tensors occupying the same coordinate description scheme.

Kinds of Object Change

Change of Direction

An active rotation applied to a vector alters its direction relative to the fixed axes while typically preserving its magnitude, so the object change in this case is confined to orientation, and any tensor built from directional information alone, such as a unit vector, is correspondingly altered in its pointing direction.

Change of Magnitude

An active scaling transformation alters the magnitude of a vector or the overall size of a higher-rank tensor's components while potentially preserving direction, producing an object change concentrated in size rather than orientation, distinguishable from a rotation by whether angles between transformed vectors are preserved.

Change of Shape for Higher-Rank Tensors

For a rank-two tensor such as a stress or strain tensor, an active transformation can alter the tensor's eigenvalues, its symmetry properties, or the orientation of its principal axes, representing a more elaborate kind of object change than a simple rotation or scaling of a single vector, since a matrix-like tensor carries more independent information than a vector.

Diagram of Object Change

Before and After the Same Fixed Frame

Effect on Invariant Quantities

Preserved Under Restricted Transformations

Some object changes preserve certain invariants even though the tensor itself is altered: an active rotation preserves vector length and preserves the determinant and trace of a rank-two tensor, while an active general linear transformation, not restricted to being orthogonal, generally does not preserve length but may still preserve other structural properties depending on the specific transformation applied.

Not Preserved in General

Unlike the passive interpretation, where a fully contracted scalar is guaranteed invariant, an active transformation offers no such blanket guarantee; whether a given scalar built from the transformed tensor equals the corresponding scalar built from the original tensor depends entirely on whether the specific active transformation applied happens to respect that particular contraction.

Sequential Active Transformations

Composition Produces Further Change

Applying a second active transformation to an already actively transformed tensor produces a third distinct object, and the net effect is described by the matrix product of the two individual transformation matrices, applied in the order matching the sequence in which the transformations were performed, since matrix multiplication for successive linear maps is generally non-commutative.

Identity as the Trivial Case

An active transformation equal to the identity matrix leaves the tensor completely unchanged, producing no object change at all, which serves as the baseline case confirming that any observed difference between an original and transformed tensor is attributable specifically to the non-identity part of the transformation applied.

Distinguishing Object Change From Passive Re-Description

The Deciding Question

Given two sets of tensor components that differ numerically, the question of whether an object change or merely a passive re-description has occurred is answered by checking whether the same basis was used for both component sets: identical basis with differing components indicates a genuine active object change, while differing bases with components related by the Jacobian and inverse Jacobian pair indicates a passive re-description of one fixed object.