1.2.41 Contravariant Tensor Definition

Contravariant tensors transform inversely to coordinate changes, preserving geometric relationships in tensor algebra.

Contravariant Tensor Definition is the characterization of a tensor built entirely from copies of a vector space itself, equivalently a tensor of type $(r, 0)$ , whose components transform under a change of basis using the inverse of the change-of-basis matrix, in contrast to and opposite from the direct transformation of basis vectors. Contravariant tensors generalize ordinary vectors to arbitrary order and form, together with covariant tensors, the two fundamental families composing the tensor algebra of a vector space.

Formal Definition

Let $V$ be a vector space over a field $F$ . A contravariant tensor of order $r$ on $V$ is an element of the tensor product space

\underset{r}{⏨} V \otimes \dots \otimes V

consisting entirely of copies of $V$ , with no factors of the dual space $V^{*}$ . By the identification of tensor product spaces with spaces of multilinear maps, a contravariant tensor of order $r$ is equivalently a multilinear map

T : V^{*} \times \dots \times V^{*} \to F

taking $r$ covector arguments and returning a scalar, linear in each argument separately.

Transformation Law

The defining feature of a contravariant tensor is how its components transform under a change of basis, and this transformation law is the source of its name. 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 contravariant tensor of order one, $T^{i}$ , transform according to

{\tilde{T}}^{i} = \sum_{j} {(A^{-1})}_{j}^{i} T^{j}

using the inverse matrix $A^{-1}$ , rather than $A$ itself. This is the origin of the term "contravariant": the components of the tensor transform inversely, or "against," the transformation used for the basis vectors, so that the underlying geometric object — the actual vector, independent of any coordinate description — remains fixed regardless of which basis is used to express it. Higher-order contravariant tensors apply this inverse matrix once for each of their $r$ indices, all written as superscripts by convention.

Why Ordinary Vectors Are Contravariant

The Coordinate Perspective

An ordinary vector $v \in V$ has coordinates $v^{i}$ relative to a basis, defined by $v = \sum_{i} v^{i} e_{i}$ . If the basis is rescaled to make basis vectors larger, the coordinates of the same fixed vector $v$ must shrink correspondingly to keep the sum $v$ unchanged; this inverse relationship between how basis vectors change and how vector coordinates change is exactly the contravariant transformation law, and it is why a type $(1, 0)$ tensor — an ordinary vector — is called contravariant.

Physical Motivation

In physical applications, quantities such as velocity, displacement, and momentum are modeled as contravariant vectors, since their numerical components must adjust inversely to a rescaling of coordinate units to describe the same underlying physical displacement or rate of change, which is precisely the behavior enforced by the contravariant transformation law.

Basic Examples by Order

Order One: Vectors

A contravariant tensor of order one is exactly an ordinary vector, an element of $V$ . Its single contravariant index reflects that it accepts one covector argument, via the natural pairing between $V$ and $V^{*}$ , and returns a scalar.

Order Two and Higher

Contravariant tensors of order two or higher generalize vectors to structures taking several covector arguments simultaneously. A symmetric contravariant tensor of order two, for instance, arises as the inverse of a metric tensor in differential geometry, used to raise indices and convert covariant quantities into contravariant ones.

Contravariant Versus Covariant Tensors

A covariant tensor is built entirely from copies of the dual space $V^{*}$ rather than $V$ , and its components transform with the matrix $A_{i}^{j}$ directly, matching the transformation of the basis vectors, rather than with its inverse. A general tensor of mixed type $(r, s)$ combines $r$ contravariant indices with $s$ covariant indices, and a purely contravariant tensor is the special case $s = 0$ . Because the two transformation laws are mutually inverse, a contravariant tensor and a covariant tensor of the same order describe genuinely different kinds of quantity unless a metric tensor is available to convert between them by raising or lowering indices.

Symmetric and Alternating Contravariant Tensors

As with covariant tensors, contravariant tensors of a fixed order $r$ subdivide according to their behavior under permutation of the arguments they accept. Symmetric contravariant tensors are unchanged under reordering of their covector arguments, while alternating contravariant tensors change sign under a single transposition; the alternating case is the natural home for multivectors, the contravariant counterparts of differential forms, used to represent oriented volumes spanned by ordinary vectors rather than by covectors.

Role Within Tensor Algebra

Contravariant tensors occupy the type- $(r, 0)$ slots of the full tensor algebra built from $V$ and its dual, generalizing ordinary vectors uniformly to arbitrary order. Because their transformation law runs inverse to the transformation of the basis, contravariant tensors describe quantities — points, displacements, velocities, and their higher-order analogues — whose intrinsic identity is independent of coordinates even though their numerical components necessarily shift with any change of basis, making the contravariant transformation law the precise algebraic statement of what it means for such a quantity to be coordinate-independent.