It has been suggested that this article or section be merged into Covariant transformation. (Discuss)

In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i.e., the property of being unchanged under some transformation. In mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry, category theory and algebraic topology. In physics they are important to the treatment of vectors and other quantities, such as tensors, that have physical meaning but are not scalars. Both special relativity (Lorentz covariance) and general relativity (general covariance) use covariant basis vectors. Wikipedia does not have an article with this exact name. ... It has been suggested that this article or section be merged with Covariant. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Theoretical physics employs mathematical models and abstractions, as opposed to experimental processes, in an attempt to understand Nature. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Table of Geometry, from the 1728 Cyclopaedia. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... ‹The template below has been proposed for deletion. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915 [1][2]. It unifies special relativity and Isaac Newtons law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time... This article or section is in need of attention from an expert on the subject. ...

In very general terms, duality interchanges covariance and contravariance, which is why these concepts occur together. For purposes of practical computation using matrices, the transpose relates two aspects (for example two sets of simultaneous equations). The case of a square matrix for which the transpose is also the inverse matrix, that is, an orthogonal matrix, is one in which covariance and contravariance can typically be treated on the same footing. This is of basic importance in the practical application of tensors. In mathematics, duality has numerous meanings. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... In mathematics, simultaneous equations are a set of equations where variables are shared. ... For the square matrix section, see square matrix. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...

A major potential cause of confusion is that this duality of covariance/contravariance intervenes every time discussion of a vector or tensor quantity is represented by its components. This causes discussion in the mathematics and physics literature often apparently to be using opposite conventions. It is not the convention that differs, but whether an intrinsic or component-wise description is the primary way of thinking of quantities. As the names suggest, covariant quantities are thought of as moving or transforming forwards, while contravariant quantities transform backwards. This depends on whether one is using a fixed background—a fact that switches the point of view.

Informal usage

In common physics usage, the adjective covariant may sometimes be used informally as a synonym for invariant (or equivariant, in mathematicians' terms). For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity; thus one might say that the Schrödinger equation is not covariant. By contrast, the Klein-Gordon equation and the Dirac equation take the same form in any coordinate frame of special relativity: thus, one might say that these equations are covariant. More properly, one should really say that the Klein-Gordon and Dirac equations are invariant, and that the Schrödinger equation is not, but this is not the dominant usage. Note also that neither the Klein-Gordon nor the Dirac equations are invariant under the transformations of general relativity (nor are they in any sense covariant either), and thus proper use should indicate what the invariance is in respect to. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... An adjective is a part of speech which modifies a noun, usually describing it or making its meaning more specific. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, is the definition of energy of a quantum system. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915 [1][2]. It unifies special relativity and Isaac Newtons law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time...

Similar informal usage is sometimes seen with respect to quantities like mass and time in general relativity: mass is technically a component of the four-momentum or the energy-momentum tensor, but one might occasionally see language referring to the covariant mass, meaning the length of the momentum four-vector. Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... A pocket watch, a device used to measure time. ... It has been suggested that this article or section be merged with Momentum#Momentum_in_relativistic_mechanics. ... The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ...

Example: covariant basis vectors in Euclidean R³

If e¹, e², e³ are contravariant basis vectors of R³ (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are: In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

Note that even if the e_i and eⁱ are not orthonormal, they are still by this definition mutually orthonormal:

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors: In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

Then v can be expressed in two (reciprocal) ways, viz.

.

The indices of covariant coordinates, vectors, and tensors are subscripts. If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts. In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

What 'contravariant' means

Contravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... For more technical Wiki articles on tensors, see the section later in this article. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

Another method is used to derive covariant tensor components. When performing tensor transformations it is critical that the method used to map to the coordinate systems in use be tracked so that operations may be applied correctly for accurate, meaningful results.

In two dimensions, for an oblique rectilinear coordinate system, contravariant coordinates of a directed line segment (in two dimensions this is termed a vector) can be established by placing the origin of the coordinate axis at the tail of the vector. Parallel lines are placed through the head of the vector. The intersection of the line parallel to the x¹ axis with the x² axis provides the x² coordinate. Similarly, the intersection of the line parallel to the x² axis with the x¹ axis provides the x¹ coordinate.

By definition, the oblique, rectilinear, contravariant coordinates of the point P above are summarized as: xⁱ = (x¹, x²) Download high resolution version (2042x1280, 56 KB)My very first Metapost creation. ...

Notice the superscript; this is a standard nomenclature convention for contravariant tensor components and should not be confused with the subscript, which is used to designate covariant tensor components.

Is there a fundamental difference in the way contravariant and covariant components can be used, or could one simply interchange them everywhere? The answer is that in curved spaces, or in curved coordinate systems in flat space (e.g. cylindrical coordinates in Euclidean space), the quantity dxⁱ is a perfect differential that can be immediately integrated to yield xⁱ, whilst the covariant components of the same differential, dx_i are not in general perfect differentials; the integrated change depends on the path. In the example of cylindrical coordinates, the radial and z components are the same in covariant and contravariant form, but the covariant component of the differential of angle round the z axis is r²dθ and its integral depends on the path. This article describes some of the common coordinate systems that appear in elementary mathematics. ...

Using the definition above, the contravariant components of a position vector vⁱ, where i = {1, 2}, can be defined as the differences between coordinates (or position vectors) of the head and tail, on the same coordinate axis. Stated in another way, the vector components are the projection onto an axis from the direction parallel to the other axis.

So, since we have placed our origin at the tail of the vector,

vⁱ = ( (x¹ − 0), (x² − 0 ) )

vⁱ = (x¹, x²)

This result is generalized into n-dimensions. Contravariance is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds. Since whether tensor components are contravariant or covariant, how they are mixed, and the order of operations all impact the results it is imperative to track for correct application of methods.

In more modern terms, the transformation properties of the covariant indices of a tensor are given by a pullback; by contrast, the transformation of the contravariant indices is given by a pushforward. This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

Use in tensor analysis

In tensor analysis, 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. For more technical Wiki articles on tensors, see the section later in this article. ...

On a manifold, a tensor field will typically have multiple indices, of two sorts. By a widely followed convention (including Wikipedia), covariant indices are written as lower indices, whereas contravariant indices are upper indices. 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 with the metric tensor. Contravariant indices can be gotten 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. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

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 as well as the cotangent bundle. In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

A contravariant vector is one which transforms like , where are the coordinates of a particle at its proper time . A covariant vector is one which transforms like , where is a scalar field.

Algebra and geometry

In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a contravariant functor. Some constructions of multilinear algebra are of 'mixed' variance, which prevents them from being functors. The distinction between homology theory and cohomology theory in topology is that homology is a covariant functor, while cohomology is a contravariant functor (it was suggested in a book, Hilton & Wylie, that contrahomology was therefore a better term for cohomology, but this did not catch on). Homology theory is covariant because (as is very clear in singular homology) its basic construction is to take a topological space X and map things into it (in that case, simplices). For a continuous mapping from X to another space Y, simply map on by composing functions. Cohomology goes the 'other way'; this is adapted to studying mappings out of X, for example the sections of a vector bundle. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... For functors in computer science, see the function object article. ... For functors in computer science, see the function object article. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... simplex refers to a one-way communications channel. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

In geometry, the same map in/map out distinction is helpful in assessing the variance of constructions. A tangent vector to a smooth manifold M is, to begin with, a curve mapping smoothly into M and passing through a given point P. It is therefore covariant, with respect to smooth mappings of M. A contravariant vector, or 1-form, is in the same way constructed from a smooth mapping from M to the real line, near P. It is in the cotangent bundle, built up from the dual spaces of the tangent spaces. Its components with respect to a local basis of one-forms dx_i will be covariant; but one-forms and differential forms in general are contravariant, in the sense that they pull back under smooth mappings. This is crucial to how they are applied; for example a differential form can be restricted to any submanifold, while this does not make the same sense for a field of tangent vectors. Table of Geometry, from the 1728 Cyclopaedia. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This article discusses the pullback in differential geometry. ... This is a glossary of terms specific to differential geometry and differential topology. ...

Covariant and contravariant components transform in different ways under coordinate transformations. By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

See also

• covariant transformation

It has been suggested that this article or section be merged with Covariant. ...

External links

• Eric W. Weisstein, Covariant Tensor at MathWorld.