10.9 Tensor Covector Component Change Rule

Tensor Covector Component Change Rule is the specific instance of the tensor component transformation law that applies to a rank-one covariant tensor, stating that the components of a covector transform under a change of basis by contraction with the forward change-of-basis matrix directly, the same matrix that carries the old basis vectors to the new basis vectors, in contrast to the components of a vector, which use the inverse of that matrix. It is the counterpart to the vector component change rule, and the two together illustrate the two fundamentally distinct ways an indexed quantity can respond to a change of basis.

Statement of the Rule

The Transformation Formula

Given a change of basis described by a forward coefficient matrix, the new components of a covector are obtained from its old components by contracting directly with that same forward matrix.

Reverse Direction of the Formula

Conversely, the old components can be recovered from the new components by contracting with the inverse matrix instead, following the inverse basis change rule applied specifically to a covector.

Why the Rule Takes This Form

Preservation of the Pairing With Vectors

A covector is defined by how it pairs with vectors to produce a scalar. For this scalar pairing to remain unchanged under a change of basis, given that vector components already transform with the inverse matrix, the covector components must transform with the forward matrix, so that the inverse and forward factors cancel when the pairing is computed.

Origin of the Term Covariant

This shared, or common, variation with the basis vectors is precisely the reason a covector's components are described as covariant: they vary together with the basis, growing when the basis vectors grow and shrinking when the basis vectors shrink, the opposite pattern from contravariant vector components.

Properties of the Rule

Linearity in the Original Components

The covector component change rule is linear in the old components, so the new components of a sum of two covectors equal the sum of the new components of each covector individually, and scaling a covector scales its new components by the same factor.

Dependence Only on the Matrix, Not the Covector

The entries of the forward matrix used in this rule depend only on the chosen change of basis and not on which particular covector is being transformed, allowing the same forward matrix to transform the components of every covector defined on the dual space.

Consistency Under Composition of Basis Changes

Applying the covector component change rule for a change from one basis to a second, followed by the rule for a change from the second basis to a third, yields the same result as applying the rule once for a direct change from the first basis to the third, with the two forward matrices composing in the corresponding order.

Schematic Representation

The diagram shows both the basis vectors and the covector components moving in the same direction under the forward matrix, illustrating the shared, covariant pattern of transformation that distinguishes a covector from a vector.

Position Within the Broader Framework

The covector component change rule is the rank-one covariant case of the general tensor component transformation law, fixing the pattern that a lower index is always paired with the forward coefficient matrix, a pattern that extends unchanged to every covariant index of a tensor of any rank.