6.20.4 Tensor Covector Transformation Pattern

Tensor Covector Transformation Pattern is the specific rule, φ'_i = A^k_i φ_k, that dictates how the components of a covector change when the underlying basis is replaced by another, using the same forward transition matrix A that relates the new basis vectors of V to the old ones. This pattern is the archetype of covariant transformation, mirroring the vector transformation pattern but using A directly rather than its inverse, and it is the second of the two most basic transformation patterns from which every higher-order tensor's transformation law is assembled.

Deriving the Pattern from Invariance of the Pairing

Setting Up the Two Bases

Let {e_i} be an original basis of V and {e'_i} a new basis related by e'_i = A^k_i e_k, with inverse B = A^{-1}. A covector φ has components φ_i = φ(e_i) in the old basis and φ'_i = φ(e'_i) in the new basis.

Substituting the Basis Relation Directly

Because φ is linear, substituting e'_i = A^k_i e_k directly into the definition of the new components gives:

producing the covector transformation pattern directly and immediately, without needing to invert any matrix, since the new components are obtained by evaluating the same fixed functional φ on the new basis vectors.

Contrast with the Indirect Derivation for Vectors

This derivation is notably more direct than the derivation of the vector transformation pattern, which required comparing two expansions of the same vector and solving for the new components using the inverse matrix B; the covector pattern falls out immediately from linearity and the definition of components as evaluations on basis vectors.

Why the Pattern Uses the Forward Matrix

Matching the Scaling of the Basis Directly

If the new basis vectors e'_i are twice as long as the old ones, then evaluating the fixed functional φ on these longer vectors produces correspondingly larger values, without any compensating inverse relationship; this direct, matching scaling is the intuitive content of the pattern and is captured precisely by the forward matrix A.

The Term Covariant

This direct relationship, in which the components vary together with the basis vectors, is the origin of the term "covariant" applied to covector components, standing in explicit contrast to the inverse, contravariant relationship exhibited by vector components.

Verifying Consistency of the Pattern

Preserving the Pairing with Vectors

Combining the covector transformation pattern with the vector transformation pattern confirms that the pairing φ(v) = φ_i v^i is invariant:

using A^k_i B^i_l = δ^k_l. This cancellation is the precise reason the two transformation patterns, one using A and the other using B, must be exactly opposite to one another: only this pairing guarantees that the scalar φ(v) does not depend on the choice of basis.

Diagram of the Covector Transformation Pattern

The Covector Pattern as the Foundation for General Covariant Behavior

Extending to Multiple Covariant Slots

The covector transformation pattern, using one factor of A, is repeated once for each additional lower index when transforming a tensor of higher covariant count, so a type (0,2) tensor uses two factors of A, one for each of its two slots, following exactly the same reasoning applied independently to each index.

The Covector Pattern as the Simplest Nontrivial Covariant Case

Because the covector transformation pattern involves only a single index and a single factor of A, it serves as the clearest setting in which to understand why covariant tensors transform the way they do, before the added complexity of multiple indices and symmetry considerations enters for higher-order covariant tensors such as the metric.