Covariance and contravariance of vectors

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. In modern mathematical notation, the role is sometimes swapped.

A simple illustrative case is that of a 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.

In contrast, a covector, also called a dual vector, has components that vary with the basis vectors in the corresponding vector space. It is an example of a covariant tensor. A covector is an object that represents a linear map from vectors to scalars. It is actually not a vector, but an object that lives in a dual vector space. Some good examples of covectors are dot product operators involving vectors. For example if ${\displaystyle \mathbf {v} }$ is a vector, then a corresponding object in the dual space would be the linear operator ${\displaystyle \mathbf {v} \ \cdot }$ . Sometimes, the components of the covector ${\displaystyle \mathbf {v} \ \cdot }$ are referred to as the covariant components of ${\displaystyle \mathbf {v} }$, although this is potentially misleading, (due to a vector having components that always vary in the contravariant sense). Despite potential confusion, this is what will be meant when the "covariant components of a vector" are referred to herein.

The gradient is often cited as an example of a covector, but this is incorrect. If the components of the gradient of a function ${\displaystyle f}$ , ${\displaystyle (\nabla f)^{i}}$ , are expressed in terms of a given basis, ${\displaystyle \mathbf {e} _{i}}$ , then these components will in fact still vary oppositely to that of the basis vectors, as can be seen by observing (using the Einstein summation convention):

where ${\displaystyle \delta _{j}^{i}}$ is the kronecker delta symbol, and ${\displaystyle (T)_{k}^{i}}$ represents the components of some transformation matrix corresponding to the transformation ${\displaystyle T}$. As can be seen, whatever transformation acts on the basis vectors, the inverse transformation must act on the components.

A third concept related to covariance and contravariance is invariance. A scalar (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis. An example of a physical observable that is a scalar is the mass of a particle. The single, scalar value of mass is independent to changes in basis vectors and consequently is called invariant. The magnitude of a vector (such as distance) is another example of an invariant, because it remains fixed even if geometrical vector components vary. (For example, for a position vector of length ${\displaystyle 3}$ meters, if all Cartesian basis vectors are changed from ${\displaystyle 1}$ meters in length to ${\displaystyle .01}$ meters in length, the length of the position vector remains unchanged at ${\displaystyle 3}$ meters, although the vector components will all increase by a factor of ${\displaystyle 100}$).

Under more general changes in basis:

• A vector or tangent vector, has components that contra-vary with a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in

  ${\displaystyle \mathbf {v} =v^{i}\mathbf {e} _{i}}$

• A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix in the corresponding (initial) vector space. The components of covectors (as opposed to those of vectors) are said to be covariant. In Einstein notation, covariant components are denoted with lower indices as in

  ${\displaystyle \mathbf {w} =w_{i}\mathbf {e} ^{i}.}$

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.

The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851 in the context of associated algebraic forms theory. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.

In the lexicon of category theory, covariance and contravariance are properties of functors; unfortunately, it is the lower-index objects (covectors) that generically have pullbacks, which are contravariant, while the upper-index objects (vectors) instead have pushforwards, which are covariant. This terminological conflict may be avoided by calling contravariant functors "cofunctors" in accordance with the "covector" terminology.