Covariance and contravariance of vectors
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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, especially in
multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...
and
tensor analysis In mathematics and physics, a tensor field is a function (mathematics), function assigning a tensor to each point of a region (mathematics), region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tens ...
, 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.
Tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s are objects in
multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...
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 In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
) of numbers such as :(v_1,v_2,v_3) . The numbers in the list depend on the choice of
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
. For instance, if the vector represents position with respect to an observer (
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
), then the coordinate system may be obtained from a system of rigid rods, or reference axes, along which the components ''v''1, ''v''2, and ''v''3 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 In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scal ...
. 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 .01 meters long), the components of the measured position
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
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 Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
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 n\times n
invertible matrix In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
''M'', so that the basis vectors transform according to \begin \mathbf_1^\prime\ \mathbf_2^\prime\ ... \ \mathbf_n^\prime\end=\begin \mathbf_1\ \mathbf_2\ ... \ \mathbf_n\endM, then the components of a vector v in the original basis ( v^i ) must be similarly transformed via \begin v^1 \\ v^2 \\ ... \\ v^n \end=M^\begin v^1 \\ v^2 \\ ... \\ v^n \end. 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 \mathbf\ \cdot , where \mathbf is a vector. The components of this covector in some arbitrary basis are \begin \mathbf\cdot\mathbf_1 & \mathbf\cdot\mathbf_2 & ... & \mathbf\cdot\mathbf_n \end, with \begin \mathbf_1\ \mathbf_2\ ... \ \mathbf_n\end 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 \mathbf , with components \begin w^1 \\ w^2 \\ ... \\ w^n \end). The covariance of these covector components is then seen by noting that if a transformation described by an n\times n
invertible matrix In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
''M'' were to be applied to the basis vectors in the corresponding vector space, \begin \mathbf_1^\prime\ \mathbf_2^\prime\ ... \ \mathbf_n^\prime\end=\begin \mathbf_1\ \mathbf_2\ ... \ \mathbf_n\endM, then the components of the covector \mathbf\ \cdot will transform with the same matrix M, namely, \begin \mathbf\cdot\mathbf_1^\prime & \mathbf\cdot\mathbf_2^\prime & ... & \mathbf\cdot\mathbf_n^\prime \end=\begin \mathbf\cdot\mathbf_1 & \mathbf\cdot\mathbf_2 & ... & \mathbf\cdot\mathbf_n \endM. 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 In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
that is a scalar is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
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 Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
) is another example of an invariant, because it remains fixed even if geometrical vector components vary. (For example, for a position vector of length 3 meters, if all Cartesian basis vectors are changed from 1 meters in length to .01 meters in length, the length of the position vector remains unchanged at 3 meters, although the vector components will all increase by a factor of 100). 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 In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...
, 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 In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
(implicit summation over repeated index), contravariant components are denoted with ''upper indices'' as in *:\mathbf = v^i \mathbf_i * A covector or
cotangent vector In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
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 In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
, covariant components are denoted with ''lower indices'' as in *:\mathbf = w_i \mathbf^i. * The scalar product of a vector and covector is the scalar v^iw_i, 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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
from R''n'' to ''V''. Regarding f as a row vector whose entries are the elements of the basis, the associated linear isomorphism is then \mathbf\mapsto \mathbf\mathbf.
Also, let the change of basis from f to f′ be given by for some
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
''n''×''n'' matrix ''A'' with entries a^i_j. Here, each vector ''Y''''j'' of the f′ basis is a linear combination of the vectors ''X''''i'' of the f basis, so that :Y_j=\sum_i a^i_jX_i, which are the columns of the matrix product \mathbf f A.


Contravariant transformation

A vector v in ''V'' is expressed uniquely as a
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of the elements X_i of the f basis as where ''v'' ''fare elements of the field ''S'' known as the components of ''v'' in the f basis. Denote the
column vector In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some , c ...
of components of ''v'' by v ''f :\mathbf mathbf= \beginv^1 mathbf\v^2 mathbf\\vdots\\v^n mathbfend so that () can be rewritten as a matrix product :v = \mathbf\, \mathbf mathbf The vector ''v'' may also be expressed in terms of the f′ basis, so that :v = \mathbf\, \mathbf mathbf However, since the vector ''v'' itself is invariant under the choice of basis, :\mathbf\, \mathbf mathbf= v = \mathbf\, \mathbf mathbf The invariance of ''v'' combined with the relationship () between f and f′ implies that :\mathbf\, \mathbf mathbf= \mathbfA\, \mathbf mathbfA giving the transformation rule :\mathbf mathbf= \mathbf mathbfA= A^\mathbf mathbf In terms of components, :v^i mathbfA= \sum_j \tilde^i_jv^j mathbf/math> where the coefficients \tilde^i_j 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: :\begin \mathbf &\longrightarrow \mathbf \\ v mathbf&\longleftarrow v mathbf\end


Covariant transformation

A
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
''α'' on ''V'' is expressed uniquely in terms of its components (elements in ''S'') in the f basis as :\alpha(X_i) = \alpha_i mathbf, \quad i=1,2,\dots,n. These components are the action of ''α'' on the basis vectors ''X''''i'' 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 :\mathbf mathbf= \begin\alpha_1 mathbf\alpha_2 mathbf\dots,\alpha_n mathbfend so that () can be rewritten as the matrix product :\alpha mathbfA= \alpha mathbf. 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: :\begin \mathbf &\longrightarrow \mathbf \\ \alpha mathbf&\longrightarrow \alpha mathbf\end Had a column vector representation been used instead, the transformation law would be the
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
:\alpha^\mathrm mathbfA= A^\mathrm\alpha^\mathrm mathbf


Coordinates

The choice of basis f on the vector space ''V'' defines uniquely a set of coordinate functions on ''V'', by means of :x^i mathbfv) = v^i mathbf The coordinates on ''V'' are therefore contravariant in the sense that :x^i mathbfA= \sum_^n \tilde^i_kx^k mathbf Conversely, a system of ''n'' quantities ''v''''i'' that transform like the coordinates ''x''''i'' 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
) on which vectors live as
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...
s or
cotangent vector In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
s. Given a local coordinate system ''x''''i'' on the manifold, the reference axes for the coordinate system are the
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s :X_1 = \frac,\dots,X_n=\frac. This gives rise to the frame at every point of the coordinate patch. If ''y''''i'' is a different coordinate system and :Y_1=\frac,\dots,Y_n = \frac, then the frame f' is related to the frame f by the inverse of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
of the coordinate transition: :\mathbf' = \mathbfJ^,\quad J = \left(\frac\right)_^n. Or, in indices, :\frac = \sum_^n\frac\frac. A tangent vector is by definition a vector that is a linear combination of the coordinate partials \partial/\partial x^i. Thus a tangent vector is defined by :v = \sum_^n v^i mathbfX_i = \mathbf\ \mathbf mathbf Such a vector is contravariant with respect to change of frame. Under changes in the coordinate system, one has :\mathbf\left mathbf'\right= \mathbf\left mathbfJ^\right= J\, \mathbf mathbf Therefore, the components of a tangent vector transform via :v^i\left mathbf'\right= \sum_^n \fracv^j mathbf Accordingly, a system of ''n'' quantities ''v''''i'' 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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' over a field ''K'' with a non-degenerate symmetric
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
(which may be referred to as the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
), there is little distinction between covariant and contravariant vectors, because the
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
allows covectors to be identified with vectors. That is, a vector ''v'' uniquely determines a covector ''α'' via :\alpha(w) = g(v, w) 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 :g(Y^i,X_j) = \delta^i_j, the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. In terms of these bases, any vector ''v'' can be written in two ways: :\begin v &= \sum_i v^i mathbf_i = \mathbf\,\mathbf mathbf\ &=\sum_i v_i mathbf^i = \mathbf^\sharp\mathbf^\sharp mathbf \end The components ''v''''i'' ''fare the contravariant components of the vector ''v'' in the basis f, and the components ''v''''i'' ''fare the covariant components of ''v'' in the basis f. The terminology is justified because under a change of basis, :\mathbf mathbfA= A^\mathbf mathbf\quad \mathbf^\sharp mathbfA= A^T\mathbf^\sharp mathbf/math> where A is an invertible n\times n matrix, and the
matrix transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
has its usual meaning.


Euclidean plane

In the Euclidean plane, the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
allows for vectors to be identified with covectors. If \mathbf_1,\mathbf_2 is a basis, then the dual basis \mathbf^1,\mathbf^2 satisfies :\begin \mathbf^1\cdot\mathbf_1 = 1, &\quad \mathbf^1\cdot\mathbf_2 = 0 \\ \mathbf^2\cdot\mathbf_1 = 0, &\quad \mathbf^2\cdot\mathbf_2 = 1. \end Thus, e1 and e2 are perpendicular to each other, as are e2 and e1, and the lengths of e1 and e2 normalized against e1 and e2, respectively.


Example

For example, suppose that we are given a basis e1, e2 consisting of a pair of vectors making a 45° angle with one another, such that e1 has length 2 and e2 has length 1. Then the dual basis vectors are given as follows: * e2 is the result of rotating e1 through an angle of 90° (where the sense is measured by assuming the pair e1, e2 to be positively oriented), and then rescaling so that holds. * e1 is the result of rotating e2 through an angle of 90°, and then rescaling so that holds. Applying these rules, we find :\mathbf^1 = \frac\mathbf_1 - \frac\mathbf_2 and :\mathbf^2 = -\frac\mathbf_1 + 2\mathbf_2. Thus the change of basis matrix in going from the original basis to the reciprocal basis is :R = \begin \frac & -\frac \\ -\frac & 2 \end, since : mathbf^1\ \mathbf^2= mathbf_1\ \mathbf_2begin \frac & -\frac \\ -\frac & 2 \end. For instance, the vector :v = \frac\mathbf_1 + 2\mathbf_2 is a vector with contravariant components :v^1 = \frac,\quad v^2 = 2. The covariant components are obtained by equating the two expressions for the vector ''v'': :v = v_1\mathbf^1 + v_2\mathbf^2 = v^1\mathbf_1 + v^2\mathbf_2 so :\begin \beginv_1\\ v_2\end &= R^\beginv^1 \\ v^2\end \\ &= \begin4 & \sqrt \\ \sqrt & 1\end \beginv^1 \\ v^2\end \\ &= \begin6 + 2\sqrt \\ 2 + \frac\end \end.


Three-dimensional Euclidean space

In the three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, one can also determine explicitly the dual basis to a given set of basis vectors e1, e2, e3 of ''E''3 that are not necessarily assumed to be orthogonal nor of unit norm. The dual basis vectors are: : \mathbf^1 = \frac ; \qquad \mathbf^2 = \frac; \qquad \mathbf^3 = \frac. Even when the ei and ei are not orthonormal, they are still mutually reciprocal: :\mathbf^i \cdot \mathbf_j = \delta^i_j, Then the contravariant components of any vector v can be obtained by the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of v with the dual basis vectors: : q^1 = \mathbf \cdot \mathbf^1; \qquad q^2 = \mathbf \cdot \mathbf^2; \qquad q^3 = \mathbf \cdot \mathbf^3 . Likewise, the covariant components of v can be obtained from the dot product of v with basis vectors, viz. : q_1 = \mathbf \cdot \mathbf_1; \qquad q_2 = \mathbf \cdot \mathbf_2; \qquad q_3 = \mathbf \cdot \mathbf_3 . Then v can be expressed in two (reciprocal) ways, viz. : \mathbf = q^i \mathbf_i = q^1 \mathbf_1 + q^2 \mathbf_2 + q^3 \mathbf_3 . or : \mathbf = q_i \mathbf^i = q_1 \mathbf^1 + q_2 \mathbf^2 + q_3 \mathbf^3 Combining the above relations, we have : \mathbf = (\mathbf \cdot \mathbf^i) \mathbf_i = (\mathbf \cdot \mathbf_i) \mathbf^i and we can convert between the basis and dual basis with :q_i = \mathbf\cdot \mathbf_i = (q^j \mathbf_j)\cdot \mathbf_i = (\mathbf_j\cdot\mathbf_i) q^j and :q^i = \mathbf\cdot \mathbf^i = (q_j \mathbf^j)\cdot \mathbf^i = (\mathbf^j\cdot\mathbf^i) q_j . If the basis vectors are orthonormal, then they are the same as the dual basis vectors.


Vector spaces of any dimension

The following applies to any vector space of dimension equipped with a non-degenerate commutative and distributive dot product, and thus also to the Euclidean spaces of any dimension. All indices in the formulas run from 1 to . The
Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
for the implicit summation of the terms with the same upstairs (contravariant) and downstairs (covariant) indices is followed. The historical and geometrical meaning of the terms ''contravariant'' and ''covariant'' will be explained at the end of this section.


Definitions

# Covariant basis of a vector space of dimension ''n'': \mathbf \triangleq , i.e. not necessarily orthonormal (D.1). # Contravariant components of a vector \mathbf: v^i \triangleq \ (D.2). # Dual (contravariant) basis of a vector space of dimension ''n'': \mathbf \triangleq \ (D.3). # Covariant components of a vector \mathbf: v_i \triangleq \ (D.4). # Components of the covariant metric tensor: g_ \triangleq \mathbf \cdot \mathbf; the metric tensor can be considered a square matrix, since it only has two covariant indices: G \triangleq \; for the commutative property of the dot product, the g_ are symmetric (D.5). # Components of the contravariant metric tensor: g^ \triangleq \; these are the elements of the inverse of the covariant metric tensor/matrix G^, and for the properties of the inverse of a symmetric matrix, they're also symmetric (D.6).


Corollaries

* g^g_ = \delta^i_k (1).
''Proof:'' from the properties of the inverse matrix (D.6). * \mathbf = g^ \mathbf (2).
''Proof:'' let's suppose that \; we will show that A^ = g^. Taking the dot product of both sides with \mathbf:
\mathbf \cdot \mathbf = (A^ \mathbf) \cdot \mathbf \stackrel A^g_ = \delta^i_k
; multiplying both sides by g^:
g^A^g_ = g^\delta^i_k \stackrel g^g_A^ = g^ \stackrel \delta^m_j A^ = g^ \to A^ = g^\stackrel g^,\ \text \quad \blacksquare
* \mathbf \cdot \mathbf = g^ (3).
''Proof:'' :
\mathbf \stackrel g^ \mathbf; \mathbf \stackrel g^ \mathbf \to \mathbf \cdot \mathbf = g^ g^ (\mathbf \cdot \mathbf) \stackrel g^g^g_ \stackrel \delta^i_m g^ = g^ \stackrel g^,\ \text \quad \blacksquare
* \mathbf = g_ \mathbf (4).
''Proof:'' let's suppose that \; we will show that B_ = g_. Taking the dot product of both sides with \mathbf: \mathbf \cdot \mathbf = (B_ \mathbf) \cdot \mathbf \stackrel B_g^ = \delta_i^k; multiplying both sides by g_:
g_B_g^ = g_\delta_i^k \stackrel g_g^B_ = g_ \stackrel \delta_m^j B_ = g_ \to B_ = g_ \stackrel g_,\ \text \quad \blacksquare
* v^i = g^ v_j (5).
''Proof:'' :
\mathbf \stackrel v^i \mathbf; \mathbf \stackrel v_j \mathbf \stackrel v_j(g^ \mathbf) \to v^i \mathbf = v_j g^ \mathbf, \forall i \to v^i = g^v_j \stackrel g^ v_j,\ \text \quad \blacksquare
* v_i = g_ v^j (6).
''Proof:'' specular to (5). * v_i = \mathbf \cdot \mathbf (7).
''Proof:'' :
\mathbf \cdot \mathbf \stackrel (v_j \mathbf) \cdot \mathbf \stackrel v_j \delta^j_i = v_i,\ \text \quad \blacksquare
* v^i = \mathbf \cdot \mathbf (8).
''Proof:'' specular to (7). * \mathbf \cdot \mathbf = g_u^iv^j (9).
''Proof:'' :
\mathbf \cdot \mathbf \stackrel (u^i\mathbf) \cdot (v^j\mathbf) = (\mathbf \cdot \mathbf)u^iv^j \stackrel g_u^iv^j,\ \text \quad \blacksquare.
* \mathbf \cdot \mathbf = g^u_iv_j (10).
''Proof:'' specular to (9).


Historical and geometrical meaning

Considering this figure for the case of an Euclidean space with n = 2, since \mathbf = \mathbf + \mathbf, if we want to express \mathbf in terms of the covariant basis, we have to multiply the basis vectors by the coefficients v^1 = \frac, v^2 = \frac. With \mathbf and thus \mathbf and \mathbf fixed, if the module of \mathbf increases, the value of the v^i component decreases, and that's why they're called ''contra''-variant (with respect to the variation of the basis vectors module). Symmetrically, corollary (7) states that the v_i components equal the dot product \mathbf \cdot \mathbf between the vector and the covariant basis vectors, and since this is directly proportional to the basis vectors module, they're called ''co''-variant. If we consider the dual (contravariant) basis, the situation is perfectly specular: the covariant components are ''contra''-variant with respect to the module of the dual basis vectors, while the contravariant components are ''co''-variant. So in the end it all boils down to a matter of convention: historically the first non- orthonormal basis of the vector space of choice was called "covariant", its dual basis "contravariant", and the corresponding components named specularly. If the covariant basis becomes orthonormal, the dual contravariant basis aligns with it and the covariant components collapse into the contravariant ones, the most familiar situation when dealing with geometrical Euclidean vectors. G and G^ become the identity matrix I, and:
g_=\delta_, g^=\delta^, \mathbf \cdot \mathbf = \delta_u^iv^j = \sum_u^iv^i = \delta^u_iv_j = \sum_u_iv_i.
If the metric is non-Euclidean, but for instance Minkowskian like in the
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
theories, the basis are never orthonormal, even in the case of special relativity where G and G^ become, for n=4,\ \eta \triangleq diag(1,-1,-1,-1). In this scenario, the covariant and contravariant components always differ.


Use in tensor analysis

The distinction between covariance and contravariance is particularly important for computations with
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s, which often have mixed variance. This means that they have both covariant and contravariant components, or both vector and covector components. The valence of a tensor is the number of covariant and contravariant terms, and in
Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
, covariant components have lower indices, while contravariant components have upper indices. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
uses more sophisticated index-free methods to represent tensors. In
tensor analysis In mathematics and physics, a tensor field is a function (mathematics), function assigning a tensor to each point of a region (mathematics), region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tens ...
, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a vector intermingle on going between covariant and contravariant expression. On a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
, a tensor field will typically have multiple, upper and lower indices, where Einstein notation is widely used. When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one another. Contravariant indices can be turned into covariant indices by
contracting A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typically involves consent to transfer of goods, services, money, or promise to transfer any of those a ...
with the metric tensor. The reverse is possible by contracting with the (matrix) inverse of the metric tensor. Note that in general, no such relation exists in spaces not endowed with a metric tensor. Furthermore, from a more abstract standpoint, a tensor is simply "there" and its components of either kind are only calculational artifacts whose values depend on the chosen coordinates. The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
as well as the cotangent bundle. A contravariant vector is one which transforms like \frac, where x^ \! are the coordinates of a particle at its
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
\tau. A covariant vector is one which transforms like \frac, where \varphi is a scalar field.


Algebra and geometry

In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, there are covariant functors and
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
s. The assignment of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
to a vector space is a standard example of a contravariant functor. Contravariant (resp. covariant) vectors are contravariant (resp. covariant) functors from a \text(n)-
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a no ...
to the fundamental representation of \text(n). Similarly, tensors of higher degree are functors with values in other representations of \text(n). However, some constructions of
multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...
are of "mixed" variance, which prevents them from being functors. In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the components of a vector relative to a basis of the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
are covariant if they change with the same linear transformation as a change of basis. They are contravariant if they change by the inverse transformation. This is sometimes a source of confusion for two distinct but related reasons. The first is that vectors whose components are covariant (called covectors or
1-form In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...
s) actually pull back under smooth functions, meaning that the operation assigning the space of covectors to a smooth manifold is actually a ''contravariant'' functor. Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of (contravariant) vectors to a smooth manifold is a ''covariant'' functor. Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather changes in the coordinate system. Vectors with contravariant components transform in the same way as changes in the coordinates (because these actually change oppositely to the induced change of basis). Likewise, vectors with covariant components transform in the opposite way as changes in the coordinates.


See also

* Active and passive transformation * Mixed tensor * Two-point tensor, a generalization allowing indices to reference multiple vector bases


Notes


Citations


References

* . * . * . * . * . * . *


External links

* * * *
Invariance, Contravariance, and Covariance
* {{tensors Tensors Differential geometry Riemannian geometry Vectors (mathematics and physics)