1.2.40 Covariant Tensor Definition

Covariant tensors transform in a specific way under coordinate changes, preserving geometric relationships in tensor calculus.

Covariant Tensor Definition is the characterization of a tensor built entirely from copies of the dual space of a vector space, equivalently a tensor of type $(0, s)$ , whose components transform under a change of basis in the same direct manner as the coefficients of a linear functional rather than the inverse manner characteristic of vector coordinates. Covariant tensors generalize linear functionals and bilinear forms to arbitrary order and constitute one of the two fundamental families, alongside contravariant tensors, into which the tensor algebra of a vector space naturally splits.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . A covariant tensor of order $s$ on $V$ is an element of the tensor product space

\underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

consisting entirely of copies of the dual space, with no factors of $V$ itself. By the identification of tensor product spaces with spaces of multilinear maps, a covariant tensor of order $s$ is equivalently a multilinear form

T : V \times \dots \times V \to F

taking $s$ vector arguments and returning a scalar, linear in each argument separately. This equivalence is the reason multilinear forms are frequently used as the working definition of covariant tensors: a covariant tensor of order $s$ and an $s$ -linear form on $V$ are simply two descriptions of the same object.

Transformation Law

The defining feature of a covariant tensor, from which its name derives, is how its components transform under a change of basis. If ${e_{i}}$ and ${{\tilde{e}}_{i}}$ are two bases of $V$ related by ${\tilde{e}}_{i} = \sum_{j} A_{i}^{j} e_{j}$ , then the components of a covariant tensor of order one, $T_{i}$ , transform according to

{\tilde{T}}_{i} = \sum_{j} A_{i}^{j} T_{j}

using the same matrix $A_{i}^{j}$ that relates the new basis vectors to the old ones directly, without inversion. This is the origin of the term "covariant": the components of the tensor transform together with, or in the same manner as, the basis vectors themselves, and higher-order covariant tensors transform by applying this same matrix once for each of their $s$ indices, all written as subscripts by convention.

Basic Examples by Order

Order One: Linear Functionals

A covariant tensor of order one is exactly a linear functional on $V$ , an element of the dual space $V^{*}$ . Its single covariant index reflects that it accepts exactly one vector argument and returns a scalar.

Order Two: Bilinear Forms

A covariant tensor of order two is a bilinear form on $V$ , with components $T_{i j}$ . Symmetric covariant tensors of order two that are additionally positive-definite are known as metric tensors, providing the algebraic foundation for measuring lengths and angles on a vector space, while antisymmetric covariant tensors of order two underlie symplectic structures.

Higher Order: General Multilinear Forms

Covariant tensors of order three or higher correspond to multilinear forms taking three or more vector arguments, appearing throughout differential geometry as curvature and torsion tensors, and in physics wherever a quantity must respond linearly and simultaneously to several vector inputs.

Covariant Versus Contravariant Tensors

A contravariant tensor is built entirely from copies of $V$ rather than $V^{*}$ , and its components transform with the inverse matrix ${(A^{-1})}^{i}$ rather than $A_{i}^{j}$ directly, matching the transformation of ordinary vector coordinates. A general tensor of mixed type $(r, s)$ combines $r$ contravariant indices with $s$ covariant indices, and a purely covariant tensor is precisely the special case $r = 0$ . Because the two transformation laws are inverse to one another, a covariant tensor and a contravariant tensor of the same order do not in general correspond to the same geometric or physical quantity unless additional structure, such as a metric tensor, is used to convert between them.

Symmetric and Alternating Covariant Tensors

Covariant tensors of a fixed order $s$ further subdivide by their behavior under permutation of arguments. Symmetric covariant tensors of order $s$ are unchanged under any reordering of their $s$ vector arguments and correspond to the symmetric multilinear forms; alternating covariant tensors of order $s$ change sign under a single transposition of arguments and correspond to the alternating multilinear forms underlying determinants and the exterior algebra. These two families, together with tensors of mixed symmetry type, exhaust the possible permutation behaviors of covariant tensors of a given order, over fields of characteristic zero.

Role Within Tensor Algebra

Covariant tensors occupy the type- $(0, s)$ slots of the full tensor algebra built from $V$ and its dual, and they are the natural home for multilinear forms of every order, generalizing linear functionals and bilinear forms uniformly to arbitrary order. Because their transformation law is the direct, matrix-forward one, covariant tensors describe quantities that respond to a change of basis in step with the basis vectors themselves, making them the natural setting for objects, such as differential forms and metric tensors, that are defined by how they act on vectors rather than by being vectors in their own right.