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. 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.
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
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 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 component of the vector in that direction would be reduced accordingly. Mathematically, if the coordinate system undergoes a transformation described by an invertible matrix M, so that the basis vectors transform according to , then the components of a vector v in the original basis ( ) must be similarly transformed via . 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 , where is a vector. The components of this covector in some arbitrary basis are , with 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 , with components ). The covariance of these covector components is then seen by noting that if a transformation described by an invertible matrix M were to be applied to the basis vectors in the corresponding vector space, , then the components of the covector will transform with the same matrix , namely, . The components of a covector are often represented arranged in a row.
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 meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at meters, although the vector components will all increase by a factor of ). The scalar product of a vector and a covector is invariant, because one has components that vary with the base change, and the other has components that vary oppositely, and the two effects cancel out. One thus says that covectors are dual to vectors.
Thus, to summarize: 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
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
The scalar product of a vector and covector is the scalar , which is invariant. It is the duality pairing of vectors and covectors.
Definition The general formulation of covariance and contravariance refers to how the components of a coordinate vector transform under a change of basis (passive transformation). Thus let V be a vector space of dimension n over a field of scalars S, and let each of and be a basis of V.A basis f may here profitably be viewed as a linear isomorphism from Rn to V. Regarding f as a row vector whose entries are the elements of the basis, the associated linear isomorphism is then Also, let the change of basis from f to f′ be given by
for some invertible n×n matrix A with entries . Here, each vector Yj of the f′ basis is a linear combination of the vectors Xi of the f basis, so that
which are the columns of the matrix product .
Contravariant transformation
A vector in V is expressed uniquely as a linear combination of the elements of the f basis as
where v[f] are elements of the field S known as the components of v in the f basis. Denote the column vector of components of v by v[f]:
so that () can be rewritten as a matrix product
The vector v may also be expressed in terms of the f′ basis, so that
However, since the vector v itself is invariant under the choice of basis,
The invariance of v combined with the relationship () between f and f′ implies that
giving the transformation rule
In terms of components,
where the coefficients are the entries of the inverse matrix of A.
Because the components of the vector v transform with the inverse of the matrix A, these components are said to transform contravariantly under a change of basis.
The way A relates the two pairs is depicted in the following informal diagram using an arrow. The reversal of the arrow indicates a contravariant change:
Covariant transformation
A linear functional α on V is expressed uniquely in terms of its components (elements in S) in the f basis as
These components are the action of α on the basis vectors Xi of the f basis.
Under the change of basis from f to f′ (via ), the components transform so that
Denote the row vector of components of α by α[f]:
so that () can be rewritten as the matrix product
Because the components of the linear functional α transform with the matrix A, these components are said to transform covariantly under a change of basis.
The way A relates the two pairs is depicted in the following informal diagram using an arrow. A covariant relationship is indicated since the arrows travel in the same direction:
Had a column vector representation been used instead, the transformation law would be the transpose
Coordinates The choice of basis f on the vector space V defines uniquely a set of coordinate functions on V, by means of
The coordinates on V are therefore contravariant in the sense that
Conversely, a system of n quantities vi that transform like the coordinates xi on V defines a contravariant vector (or simply vector). A system of n quantities that transform oppositely to the coordinates is then a covariant vector (or covector).
This formulation of contravariance and covariance is often more natural in applications in which there is a coordinate space (a manifold) on which vectors live as tangent vectors or cotangent vectors. Given a local coordinate system xi on the manifold, the reference axes for the coordinate system are the vector fields
This gives rise to the frame at every point of the coordinate patch.
If yi is a different coordinate system and
then the frame f' is related to the frame f by the inverse of the Jacobian matrix of the coordinate transition:
Or, in indices,
A tangent vector is by definition a vector that is a linear combination of the coordinate partials . Thus a tangent vector is defined by
Such a vector is contravariant with respect to change of frame. Under changes in the coordinate system, one has
Therefore, the components of a tangent vector transform via
Accordingly, a system of n quantities vi depending on the coordinates that transform in this way on passing from one coordinate system to another is called a contravariant vector.
Covariant and contravariant components of a vector with a metric
In a finite-dimensional vector space V over a field K with a non-degenerate symmetric bilinear form (which may be referred to as the metric tensor), there is little distinction between covariant and contravariant vectors, because the bilinear form allows covectors to be identified with vectors. That is, a vector v uniquely determines a covector α via
for all vectors w. Conversely, each covector α determines a unique vector v by this equation. Because of this identification of vectors with covectors, one may speak of the covariant components or contravariant components of a vector, that is, they are just representations of the same vector using the reciprocal basis.
Given a basis of V, there is a unique reciprocal basis of V determined by requiring that
the Kronecker delta. In terms of these bases, any vector v can be written in two ways:
The components vi[f] are the contravariant components of the vector v in the basis f, and the components vi[f] are the covariant components of v in the basis f. The terminology is justified because under a change of basis,
where is an invertible matrix, and the matrix transpose has its usual meaning.
Euclidean plane