1.2.61 Tensor Covector Component Definition

In tensor algebra, a covector component defines how a covector acts on vectors, essential for understanding dual spaces and tensor transformations.

Tensor Covector Component Definition is the specification of the individual numerical entries that make up the component array of a rank-one, type (0, 1) tensor, that is, a tensor with a single lower index and no upper indices, identifying each entry as the coefficient of one particular dual basis covector in the expansion of a linear functional relative to a chosen basis of the underlying vector space. A tensor covector component is the covariant counterpart of a tensor vector component, arising from the dual space rather than the vector space itself, and it is the building block from which the coordinate representation of a linear functional is assembled.

Definition via Dual Basis Expansion

The Covariant Vector

A type (0, 1) tensor ω is an element of the dual space V*, also called a covariant vector, a covector, or a one-form. Given a basis e_1, ..., e_n of V and its associated dual basis e^1, ..., e^n of V*, the covector ω can be written uniquely as a linear combination of the dual basis covectors.

ω = ω_{i} e^{i} = \sum_{i = 1}^{n} ω_{i} e^{i}

The coefficients ω_1, ω_2, ..., ω_n appearing in this expansion are the tensor covector components of ω relative to the chosen basis, each one a single element of the underlying field.

Single Lower Index

Because a type (0, 1) tensor has exactly one covariant slot and no contravariant slots, its components carry exactly one lower index, ω_i, and no upper index, distinguishing covector components from vector components, which carry a single upper index instead.

Extraction of a Component

Using the Basis of V Directly

Since ω is a linear functional on V, each covector component can be recovered by evaluating ω directly on the corresponding basis vector of V.

ω_{i} = ω (e_{i})

This mirrors the way a vector component is obtained by pairing the vector with the dual basis, except here the roles are exchanged: the covector itself acts on the vector space basis.

Transformation of Covector Components

Covariant Transformation Law

Under a change of basis described by a matrix A, relating a new basis of V to the old one, the covector components transform using the inverse matrix A^{-1}, which is the origin of the term "covariant": the components transform in the same direction as the basis vectors of V themselves, unlike vector components, which transform oppositely.

{\tilde{ω}}_{l} = {(A^{- 1})}_{l}^{j} ω_{j}

Invariance of the Covector Itself

Although the numerical values ω_1, ..., ω_n change when the basis changes, the combination ω_i e^i remains equal to the same abstract covector ω in every basis, because the transformation applied to the components exactly compensates for the transformation applied to the dual basis covectors.

Action on a Vector

Pairing a Covector with a Vector

Given a vector v with components v^i and a covector ω with components ω_i in the same basis, the action of ω on v, producing a scalar, is computed by summing the products of corresponding components.

ω (v) = ω_{i} v^{i}

This scalar result is basis-independent even though both the covector components and the vector components individually depend on the basis, because the opposite transformation behaviors of upper and lower indices exactly cancel in the summation.

Geometric Interpretation

Level Sets Instead of Arrows

While a vector component describes a coordinate along a basis direction and is naturally pictured as an arrow, a covector is more naturally pictured through its level sets, the family of parallel hyperplanes on which the covector takes a constant value; the covector components describe how densely these level sets are packed along each basis direction.

Examples of Covectors

Common examples represented as type (0, 1) tensors include the differential of a scalar function, the gradient expressed in covariant form, and any linear measuring device that assigns a single number to each vector, such as a component of momentum conjugate to a coordinate.

Special Values

Zero Covector

A covector all of whose components equal zero in some basis has all components equal to zero in every other basis as well, and it assigns the value zero to every vector in V.

Dual Basis Covector Components

The i-th dual basis covector e^i itself has components that equal 1 in position i and 0 in every other position, the same Kronecker delta pattern seen for standard basis vectors, ω_k = δ_{ki}.