In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. ^[2] Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851.^[3][4]

Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated with any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the coordinate description of a vector changes by passing from one coordinate system to another. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.

Introduction

In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or tuple) of numbers such as

${\displaystyle (v_{1},v_{2},v_{3}).}$

The numbers in the list depend on the choice of coordinate system. For instance, if the vector represents position with respect to an observer (position vector), then the coordinate system may be obtained from a system of rigid rods, or reference axes, along which the components v₁, v₂, and v₃ are measured. For a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system. That is to say, the components of the vectors will transform in a certain way in passing from one coordinate system to another.

A simple illustrative case is that of a Euclidean vector. For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary opposite to that of the basis vectors. That vector is therefore defined as a contravariant tensor. Take a standard position vector for example. By changing the scale of the reference axes from meters to centimeters (that is, dividing the scale of the reference axes by 100, so that the basis vectors now are ${\displaystyle .01}$ meters long), the components of the measured position vector are multiplied by 100. A vector's components change scale inversely to changes in scale to the reference axes, and consequently a vector is called a contravariant tensor.

A vector, which is an example of a contravariant tensor, has components that transform inversely to the transformation of the reference axes, (with example transformations including rotation and dilation). The vector itself does not change under these operations; instead, the components of the vector change in a way that cancels the change in the spatial axes. In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, would reduce in an exactly compensating way. Mathematically, if the coordinate system undergoes a transformation described by an ${\displaystyle n\times n}$ invertible matrix M, so that the basis vectors transform according to ${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}^{\prime }\ \mathbf {e} _{2}^{\prime }\ ...\ \mathbf {e} _{n}^{\prime }\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\ \mathbf {e} _{2}\ ...\ \mathbf {e} _{n}\end{bmatrix}}M}$, then the components of a vector v in the original basis ( ${\displaystyle v^{i}}$ ) must be similarly transformed via ${\displaystyle {\begin{bmatrix}v^{1}{^{\prime }}\\v^{2}{^{\prime }}\\...\\v^{n}{^{\prime }}\end{bmatrix}}=M^{-1}{\begin{bmatrix}v^{1}\\v^{2}\\...\\v^{n}\end{bmatrix}}}$. The components of a vector are often represented arranged in a column.

By contrast, a covector has components that transform like the reference axes. It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector. To illustrate, assume we have a covector defined as ${\displaystyle \mathbf {v} \ \cdot }$ , where ${\displaystyle \mathbf {v} }$ is a vector. The components of this covector in some arbitrary basis are ${\displaystyle {\begin{bmatrix}\mathbf {v} \cdot \mathbf {e} _{1}&\mathbf {v} \cdot \mathbf {e} _{2}&...&\mathbf {v} \cdot \mathbf {e} _{n}\end{bmatrix}}}$, with ${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}\ \mathbf {e} _{2}\ ...\ \mathbf {e} _{n}\end{bmatrix}}}$ being the basis vectors in the corresponding vector space. (This can be derived by noting that we want to get the correct answer for the dot product operation when multiplying by an arbitrary vector ${\displaystyle \mathbf {w} }$ , with components ${\displaystyle {\begin{bmatrix}w^{1}\\w^{2}\\...\\w^{n}\end{bmatrix}}}$). The covariance of these covector components is then easily seen by noting that if a transformation described by an ${\displaystyle n\times n}$ invertible matrix M were to be applied to the basis vectors in the corresponding vector space, ${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}^{\prime }\ \mathbf {e} _{2}^{\prime }\ ...\ \mathbf {e} _{n}^{\prime }\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\ \mathbf {e} _{2}\ ...\ \mathbf {e} _{n}\end{bmatrix}}M}$, then the components of the covector ${\displaystyle \mathbf {v} \ \cdot }$ will transform with the same matrix ${\displaystyle M}$, namely,