9.16 Tensor Basis Dependent Component Behavior

Tensor Basis Dependent Component Behavior is the general pattern by which the numerical values of a tensor's components change in a predictable, rule-governed way whenever the underlying basis is changed, while the tensor itself remains fixed. It describes the overall behavior exhibited by components as a class, rather than any single transformation rule in isolation.

The Core Behavior

Components Are Not Fixed Numbers

Unlike a plain scalar, whose value does not depend on any choice of basis, the components of a vector, covector, or higher-order tensor are only meaningful relative to a specific basis, and they take different numerical values under different bases even though they describe the same underlying object.

Predictability of the Change

Although components change numerically under a change of basis, they do not change arbitrarily. The new components are always determined completely by the old components and the transformation relating the old basis to the new one, following a fixed rule that depends only on the type of the tensor.

Categories of Behavior

Contravariant Behavior

Components associated with upper indices exhibit contravariant behavior, transforming with the inverse of the matrix that transforms the basis vectors, so that they vary oppositely to the basis in order to keep the tensor itself fixed.

Covariant Behavior

Components associated with lower indices exhibit covariant behavior, transforming with the same matrix that transforms the basis vectors, so that they vary together with the basis.

Mixed Behavior

Tensors carrying both upper and lower indices exhibit a combination of contravariant and covariant behavior, with each index transforming according to its own type independently of the others, so that the overall transformation of a mixed tensor's components is the product of the individual index transformations.

What Remains Invariant

Full Contractions

Even though individual components vary under basis change, quantities obtained by fully contracting all indices of a tensor, or of several tensors together, remain numerically unchanged, since the contravariant and covariant transformations cancel exactly in such contractions.

The Tensor Itself

The most fundamental invariant is the tensor itself: no matter how its components behave numerically under a change of basis, the abstract object being represented, reconstructed through the summation form, is always the same.

Implications of This Behavior

Why Basis Must Always Be Specified

Because components depend on the basis, any statement of a tensor's numerical component values is incomplete unless the basis relative to which those values were computed is also specified. Reporting components without their associated basis leaves the description ambiguous.

Guiding Correct Manipulation

Understanding basis dependent behavior is what allows one to determine, for any given tensor operation, whether its numerical result will remain the same or will change under a change of basis, and consequently whether that operation produces a basis-independent, tensorial quantity or a quantity that is only meaningful relative to a particular basis.