NationMaster - Encyclopedia: Covariance and contravariance of vectors

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For other uses of "covariant" or "contravariant", see covariance and contravariance.

1 Definition
2 Context
3 Informal usage: invariance
4 Rules of Covariant and Contravariant Transformation
5 Dual basis
- 5.1 Euclidean R3
6 What 'contravariant' means
7 Use in tensor analysis
8 Algebra and geometry
9 See also
10 External links

Image File history File links Broom_icon. ... Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged with Covariant. ... It has been suggested that this article or section be merged into Covariant transformation. ...

Definition

In mathematics and theoretical physics, covariance and contravariance refer to how coordinates change under a transformation. Components of vectors transform contravariantly, while components of covectors transform covariantly (cov- with cov-). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... Look up vector in Wiktionary, the free dictionary. ... A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

The distinction is particularly important for computations with tensors, which often have mixed variance (both covariant and contravariant components). Using Einstein notation, covariant components have lower indices, while contravariant components have upper indices. 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. ... This article or section does not adequately cite its references or sources. ...

When one chooses coordinates on a vector space V, for concreteness say Euclidean n-space $mathbf{R}^n$ , both vectors and covectors can be written as an n-tuple of numbers, $(x_1,dots,x_n)$ , but if one changes the basis, they transform differently. Vectors are called contravariant vectors, while covectors are called covariant vectors.

Given a basis for a vector space, a transform $Tcolon V to V$ is represented by a matrix M, while the dual transform $T^*colon V^* leftarrow V^*$ is represented by the transpose $M t$ , and the inverse dual transform $T^*colon V^* to V^*$ is represented by the inverse of the transpose $(M t) - 1$ (equivalently, transpose of the inverse); duality reverses direction (it is a contravariant functor), hence the need for the inverse to reverse direction. Thus vectors transform as M, while covectors transform via $(M t) - 1$ . In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... For functors in computer science, see the function object article. ...

These matrices agree if and only if M is an orthogonal matrix, in which case covariant and contravariant vectors transform identically. 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. ...

Context

One can contrast covariance and contravariance (transforming in a particular way) with invariance, i.e., the property of being unchanged under some transformation. In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ...

Both special relativity (Lorentz covariance) and general relativity (general covariance) use covariant basis vectors. 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... â€¹The template below has been proposed for deletion. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... This article or section is in need of attention from an expert on the subject. ...

Systems of simultaneous equations are contravariant in the variables. This article or section is in need of attention from an expert on mathematics. ...

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: invariance

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 Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... talea harris and sophie king are sluts In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun (called the adjectives subject, giving more information about what the noun or pronoun refers to. ... 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... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... 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 SchrÃ¶dinger equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

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. Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... 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. ...

Rules of Covariant and Contravariant Transformation

Vectors are covariant, and covectors are contravariant, but the components of vectors are contravariant and the components of covectors are covariant. See Einstein notation for details. This article or section does not adequately cite its references or sources. ...

This is a frequently confused point.

In tensor representation a vector $mathbf{A}$ can be expressed as the sum of the products of each of its components times the basis vector belonging to that component in two ways (repeated indices are assumed to sum according to the Einstein summation convention): For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

$mathbf{A}=a^i mathbf{e}_i=a_i mathbf{e}^i$

where $a i$ are called the contravariant components of $mathbf{A}$ , $a i$ are called the covariant components of $mathbf{A}$ , $mathbf{e}_i$ are covariant basis vectors, and $mathbf{e}^i$ are contravariant basis vectors if and only if these transform from coordinates $x' i$ to coordinates $x i$ (where $x i$ are differentiable functions of $x' i$ , and vice versa) according to the rules:

$a^i=a'^j {partial x^i over partial x'^j }$ ,

$a_i=a'_j {partial x'^j over partial x^i }$ ,

$mathbf{e}^i=mathbf{e'}^j {partial x^i over partial x'^j }$ ,

$mathbf{e}_i=mathbf{e'}_j {partial x'^j over partial x^i }$ ,

where the primed components and basis vectors represent A in the coordinates $x' i$ :

$mathbf{A}=a'^i mathbf{e'}_i=a'_i mathbf{e'}^i$ .

We could also compute the inverse relations:

$a'^i=a^j {partial x'^i over partial x^j }$ ,

$a'_i=a_j {partial x^j over partial x'^i }$ ,

$mathbf{e'}^i=mathbf{e}^j {partial x'^i over partial x^j }$ ,

$mathbf{e'}_i=mathbf{e}_j {partial x^j over partial x'^i }$ ,

which is only possible if the determinant of the matrices formed by the components of $partial x^i / partial x'^j$ and $partial x'^j / partial x^i$ are non-zero. The determinant of the matrix formed by $partial x^i / partial x'^j$ is called the Jacobian $J$ of the transformation, which must be non-zero to provide a complete set of transformation laws. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

Note that the matrices formed by all of the above partial derivative transformations can be generated as the inverse, transpose, and transpose of the inverse of the matrix formed by the components of $partial x^i / partial x'^j$ . The key property of the tensor representation is the preservation of invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner), and these operations are inverse to one another according to the transformation rules. Substituting the transformation rules for the definition of $mathbf{A}$ gives:

$mathbf{A}=a^i mathbf{e}_i=a'_j {partial x'^j over partial x^i }mathbf{e'}^j {partial x^i over partial x'^j }=a'_jmathbf{e'}^j$

where the partial derivative terms cancel one another since they must be inverse to one another. This illustrates what is meant by invariance. A similar relation holds for all vectors (or higher-order tensors), allowing them to be written in the manner described above. Using the transformation rules can also show that: $mathbf{e}^icdotmathbf{e}_j =delta^i_j$ , where $delta^i_j$ is 1 if $i = j$ and 0 otherwise.

Note that in this kind of system the basis vectors are not generally of unit length, nor are covariant basis vectors necessarily parallel to their contravariant basis vectors (if the coordinates are non-orthogonal).

Illustration of the contravariant and covariant representation of vectors in a 2D curvilinear, non-orthogonal grid

The above figure illustrates how the contravariant and covariant representations would be plotted in terms of components on a 2D curvilinear non-orthogonal grid for a generic vector $mathbf{A}$ . Note that the sum of either pair of vectors yields the same vector. Also note that the covariant basis vectors are parallel to their respective coordinate lines while the contravariant basis vectors are orthogonal to the directions of the other coordinate lines.

There are many other useful properties of the tensor representation. If we take the dot product of $mathbf{A}=a^i mathbf{e}_i=a_k mathbf{e}^k$ and $mathbf{e}_j$ then we obtain:

$a_j=a^i g_{ij}=a^i(mathbf{e}_icdot mathbf{e}_j)$

where $g i j$ is the covariant metric tensor. The dot product of $mathbf{A}=a^k mathbf{e}_k=a_j mathbf{e}^j$ and $mathbf{e}^i$ likewise gives: In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

$a^i=a_j g^{ij}=a_j(mathbf{e}^icdot mathbf{e}^j)$

where $g i j$ is the contravariant metric tensor. This gives two useful results: 1) the covariant (or contravariant) components of a vector can be recovered by taking the dot product of that vector and the covariant (or contravariant) basis vectors, and 2) the covariant and contravariant components are related by the metric tensor. We note in passing that the covariant and contravariant basis vectors are also related to one another by the metric tensor, and that the above relations require that $g i j$ and $g i j$ are inverse to one another. In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...

We note that the tensor representation is not restricted to vectors, but can be used on higher-order tensors where each covariant or contravariant component transforms individually according to the rules described above. For example, we could transform a so-called mixed tensor of the form: In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ...

$b^i_j = {b'}^k_l {partial x^i over partial x'^k} {partial x'^l over partial x^j}$

by successively applying the transformation rules to each index according to whether it is covariant (lowered) or contravariant (raised).

Dual basis

Given a basis $e_1, dots, e_n$ of a vector space V, there is a unique dual basis $e^1, dots, e^n$ of the dual space, which is determined by requiring In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...

$mathbf{e}^i cdot mathbf{e}_j = delta^i_j$ .

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...

$mathbf{e}_1 = frac{mathbf{e}^2 times mathbf{e}^3}{mathbf{e}^1 cdot (mathbf{e}^2 times mathbf{e}^3)} ; qquad mathbf{e}_2 = frac{mathbf{e}^3 times mathbf{e}^1}{mathbf{e}^2 cdot (mathbf{e}^3 times mathbf{e}^1)}; qquad mathbf{e}_3 = frac{mathbf{e}^1 times mathbf{e}^2}{mathbf{e}^3 cdot (mathbf{e}^1 times mathbf{e}^2)}.$

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

$mathbf{e}^i cdot mathbf{e}_j = delta^i_j$ .

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 over the real numbers R and returns a real-valued scalar quantity. ...

$q^1 = mathbf{v} cdot mathbf{e}^1; qquad q^2 = mathbf{v} cdot mathbf{e}^2; qquad q^3 = mathbf{v} cdot mathbf{e}^3$ .

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

$q_1 = mathbf{v} cdot mathbf{e}_1; qquad q_2 = mathbf{v} cdot mathbf{e}_2; qquad q_3 = mathbf{v} cdot mathbf{e}_3$ .

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

$mathbf{v} = q_i mathbf{e}^i = q_1 mathbf{e}^1 + q_2 mathbf{e}^2 + q_3 mathbf{e}^3$

$mathbf{v} = q^i mathbf{e}_i = q^1 mathbf{e}_1 + q^2 mathbf{e}_2 + q^3 mathbf{e}_3$ .

Combining the above relations, we have

$mathbf{v} = (mathbf{v} cdot mathbf{e}_i) mathbf{e}^i = (mathbf{v} cdot mathbf{e}^i) mathbf{e}_i$

and we can convert from covariant to contravariant basis with

$q_i = mathbf{v}cdot mathbf{e}_i = (q^j mathbf{e}_j)cdot mathbf{e}_i = (mathbf{e}_jcdotmathbf{e}_i) q^j$

and

$q^i = mathbf{v}cdot mathbf{e}^i = (q_j mathbf{e}^j)cdot mathbf{e}^i = (mathbf{e}^jcdotmathbf{e}^i) q_j$ .

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 set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

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 (differential). Suppose that Ï†:Mâ†’ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is... Suppose that Ï† : M â†’ N is a smooth map between smooth manifolds; then the differential of Ï† at a point x is, in some sense, the best linear approximation of Ï† near x. ...

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 180Â°. 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 âˆˆ M and v âˆˆ 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 $frac{dx^{mu}}{dtau}$ , where $x^{mu} !$ are the coordinates of a particle at its proper time $tau !$ . A covariant vector is one which transforms like $frac{partial phi}{partial x^{mu}}$ , where $phi !$ 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. In mathematics, category theory 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 (1Ã—n) and column vectors (nÃ—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. ... Look up section in Wiktionary, the free dictionary. ... 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. Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... 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 (1Ã—n) and column vectors (nÃ—1). ... The tangent space of a manifold is a concept which facilitates the generalization of 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. ... Suppose that Ï†:Mâ†’ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is... 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, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ... Suppose that Ï†:Mâ†’ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...

External links

Eric W. Weisstein, Covariant Tensor at MathWorld.

Categories: Cleanup from March 2007 | All pages needing cleanup | Articles to be merged since October 2006 | Tensors | Differential topology | Riemannian geometry Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

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