7.15 Tensor Component Change Behavior

Tensor Component Change Behavior is the general study of how a tensor's numerical entries respond when the basis used to express them is altered, encompassing the distinct transformation rules that govern contravariant, covariant, and mixed indices as the frame of reference is changed.

The Central Fact of Change Behavior

Components Are Not Fixed Numbers

A tensor's components are never absolute, unchanging numbers; they are numbers relative to a chosen basis, and the defining feature of a tensor, as opposed to an arbitrary array of numbers, is that its components change in a precise, predictable way whenever the basis changes.

Predictability as the Defining Trait

What distinguishes a tensor's change behavior from arbitrary numerical variation is that the new components can always be computed directly from the old components together with the transition matrix connecting the two bases, without needing to return to the tensor's original abstract definition.

The Two Fundamental Change Behaviors

Contravariant Change Behavior

Contravariant components, marked by upper indices, change using the inverse of the transition matrix relating the new basis to the old, meaning they respond to a change of basis in the opposite sense to the basis vectors themselves.

Covariant Change Behavior

Covariant components, marked by lower indices, change using the transition matrix directly, meaning they respond to a change of basis in the same sense as the basis vectors, which is the origin of the term covariant.

Combined Change Behavior for Higher-Rank Tensors

One Factor Per Index

A tensor carrying several indices exhibits a change behavior built by combining one transformation factor for each index it possesses, applying the contravariant rule to every upper index and the covariant rule to every lower index within the same overall expression.

Consistency Across the Whole Tensor

Every index changes according to its own individually assigned rule regardless of how many other indices the tensor carries, so the overall change behavior of a complicated tensor is entirely predictable once the variance type of each index is known.

Quantities Exempt From Change

Invariants Under the General Change Behavior

Certain specific combinations of components, formed by fully contracting every upper index against a matching lower index, remain unchanged regardless of the basis, since the contravariant and covariant transformation factors cancel exactly in such combinations.

The Purpose of Studying Change Behavior

Understanding exactly how components change is what allows these invariant combinations to be identified in the first place, since recognizing an invariant requires knowing precisely how the transformation factors for each index interact and potentially cancel.

Diagrammatic Illustration

Contravariant and covariant components responding in opposite directions to the same rescaling of a basis vector.

Broader Significance of Change Behavior

Foundation of the Tensor Concept Itself

The precise, rule-governed change behavior of components under a change of basis is, in essence, what makes a tensor a tensor: any array of numbers that fails to obey one of the recognized change behaviors, contravariant, covariant, or a consistent combination thereof, does not qualify as a tensor at all.

Necessity for Physical and Geometric Consistency

In physical and geometric applications, correct change behavior ensures that equations expressed in tensor form remain valid regardless of the coordinate system used to describe them, which is precisely the property that makes tensor formulations so valuable for expressing laws that must hold true independent of any particular observer's chosen frame of reference.