10.8.2 Tensor Vector Component Contravariant Behavior

Tensor vector components change inversely to coordinate system scaling, reflecting contravariant behavior under linear transformations.

Tensor Vector Component Contravariant Behavior is the qualitative pattern, following directly from the vector component change rule, in which a vector's components scale inversely to the scaling of the basis vectors under a change of basis, so that stretching the basis vectors causes the components to shrink and shrinking the basis vectors causes the components to grow, while the vector itself remains unchanged. It is the conceptual counterpart to the formal transformation formula, describing in qualitative terms why the components of a vector are said to vary contrary to the variation of the basis.

The Behavior Described

Inverse Scaling Relationship

When a basis vector is rescaled by some factor, the component of a vector along that basis vector must be rescaled by the reciprocal of that factor, so that the product of the component and the basis vector, which represents the actual contribution to the vector, remains the same.

v^{i} e_{i} = (\frac{1}{λ} v^{i}) (λ e_{i})

This simple scalar case, where the basis is stretched by a single factor, illustrates the general principle: the component adjusts by the reciprocal factor to keep the total contribution unchanged.

General Case Through the Inverse Matrix

For a general change of basis, not merely a uniform rescaling, the same inverse relationship is captured precisely by the inverse coefficient matrix appearing in the full vector component change rule, generalizing the simple reciprocal scaling to a full linear transformation.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Origin of the Term Contravariant

Contrary Variation

The word contravariant describes components that vary contrary to, that is, in the opposite sense from, the basis vectors they accompany. This contrasts with covariant components, which vary in the same sense as the basis vectors, using the forward matrix rather than its inverse.

Necessity for Invariance

This contrary variation is not an arbitrary labeling choice; it is forced by the requirement that the vector, reconstructed from its components and the basis vectors, remain invariant under the change of basis. Without the contrary scaling of the components, the reconstructed vector would change whenever the basis changed, contradicting the idea that a vector is a fixed geometric or algebraic object.

Illustrating the Behavior

Uniform Stretching Example

If every basis vector is stretched by the same factor, every component of every vector expressed in that basis shrinks by exactly that same factor, leaving the represented vectors themselves completely unchanged in magnitude and direction.

Non-Uniform Change Example

If the basis vectors are stretched by different amounts, or rotated relative to one another, the components no longer scale by a single common reciprocal factor, but each component still adjusts precisely according to the inverse matrix so that the reconstructed vector remains fixed.

Schematic Representation

The diagram compares a short basis vector requiring a large component to reach the same total extent as a long basis vector requiring only a small component, illustrating the inverse relationship at the heart of contravariant behavior.

Role Within the Broader Framework

Tensor Vector Component Contravariant Behavior is the qualitative description underlying the vector component change rule and its associated change matrix, and it generalizes directly to every upper index of a tensor of arbitrary rank, each of which exhibits the same contrary variation relative to the basis vectors of the corresponding vector space.