NationMaster - Encyclopedia: Covector

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. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

1 Introduction
2 Visualizing one-forms
3 Basis of the dual space
4 Differential one-forms
5 See also
6 Reference

Introduction

A one-form is a tensor of type $begin{pmatrix} 0 1 end{pmatrix}$ . It is the simplest non-scalar tensor. In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ...

Let $tilde{f}$ represent a one-form which acts on vectors of space V, including vectors $vec u$ and $vec v$ . Then the linearity properties of $tilde{f}$ are The word linear comes from the Latin word linearis, which means created by lines. ...

$tilde{f} (vec u + vec v) = tilde{f} (vec u) + tilde{f} (vec v)$

$tilde{f} (alpha vec v) = alpha tilde{f} (vec v)$

where α is a scalar. The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ...

The set of all one-forms definable on the vector space V can also itself be a vector space if one-forms can be added to each other or be multiplied by scalars in a pointwise linear manner. That is, if the vectors of the space V are position vectors of points, then for every point $vec v$ in the space V, the following should hold true:

$(tilde{f} + tilde{g}) (vec v) = tilde{f}(vec v) + tilde{g}(vec v)$

$(alpha tilde{f}) (vec v) = alpha tilde{f}(vec v).$

If these last two conditions are true for every $vec v isin V$ then the one-forms constitute a vector space.

If V is an inner-product space with inner product 〈 , 〉 then every vector $vec v$ can be mapped to a dual one-form $tilde{v}$ defined by In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

$tilde{v} := langle vec v, rangle$

(i.e. $tilde{v} := lambda x. langle vec v, x rangle$ in lambda notation) so that the one-form $tilde{v}$ applied to a vector $vec u$ yields The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...

$tilde{v} (vec u) = langle vec v, vec urangle.$

Thus the inner product provides a bijection of each vector in V to a one-form of its dual vector space $tilde{V}$ . In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

Visualizing one-forms

A vector is usually visualized as an arrow extending from the origin to a point in space. A one-form can be visualized as a set of equally spaced parallel planes which partition the entire space. The magnitude of a one-form is directly proportional to the density of parallel planes and inversely proportional to the spacing between pairs of neighboring planes. To find the result of applying a one-form to a vector, basically count the number of planes which a vector cuts through. (Note: this visualization is discrete whereas one-forms and vectors have magnitudes which range continuously over the real numbers. The visualization can be interpolated linearly, as it were, to increase the precision.)

Basis of the dual space

Let the vector space V have a basis ${vec e}_1, {vec e}_2$ , … , ${vec e}_n$ , not necessarily orthonormal nor even orthogonal. Then the dual space $tilde{V}$ has a basis $tilde{omega}^1, tilde{omega}^2$ , … , $tilde{omega}^n$ which in the three-dimensional case (n = 3) can be defined by 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. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... 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). ...

$tilde{omega}^i = {1 over 2} , leftlangle { epsilon^{ijk} , (vec e_j times vec e_k) over vec e_1 cdot vec e_2 times vec e_3} , qquad rightrangle$

where $epsilon,!$ is the Levi-Civita symbol . This definition has the special property that In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is defined as follows: i. ...

$tilde{omega}^i (vec e_j) = delta^i {}_j$

where δ is the Kronecker delta. Thus, these two dual bases are mutually orthonormal even if each basis is not self-orthonormal. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

N.B. The superscripts of the basis one-forms are not exponents but are instead contravariant indices. This page does not deal with the statistical concept covariance of random variables, nor with the computer science concepts of covariance and contravariance. ...

A one-form $tilde{u}$ belonging to the dual space $tilde{V}$ can be expressed as a linear combination of basis one-forms, with coefficients ("components") u_i , In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

$tilde{u} = u_i , tilde{omega}^i$

Then, applying one-form $tilde{u}$ to a basis vector e_j yields

$tilde{u}(vec e_j) = (u_i , tilde{omega}^i) vec e_j = u_i (tilde{omega}^i (vec e_j))$

due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then

$tilde{u}({vec e}_j) = u_i (tilde{omega}^i ({vec e}_j)) = u_i delta^i {}_j = u_j$

that is

$tilde{u} (vec e_j) = u_j.$

This last equation shows that an individual component of a one-form can be extracted by applying the one-form to a corresponding basis vector.

Differential one-forms

A differential one-form is a one-form the components of which are all differential. It is the simplest non-scalar differential form. A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Reference

Bernard F. Schutz (1985, 2002). A first course in general relativity. Cambridge University Press: Cambridge, UK. Chapter 3. ISBN 0-521-27703-5.

Categories: Multivariate calculus | Tensors The International Standard Book Number, or ISBN (sometimes pronounced is-ben), is a unique identifier for books, intended to be used commercially. ...

Results from FactBites:

PlanetMath: covector (73 words)
	Thus, for example, a covector field on a differentiable manifold is a synonym for a 1-form.
	This is version 3 of covector, born on 2002-12-10, modified 2003-09-16.
	Object id is 3719, canonical name is Covector.

Math Forum - Ask Dr. Math (639 words)
	Another way to define a covector is that it's a linear function that takes each vector to a number: in other words, you can multiply a covector by any vector and get a number.
	If you "feed" it only a covector, multiplying cM, you're left with a covector, still "hungry" for a vector to produce a final numerical answer.
	And if you "feed" it less than its full complement of desired vectors and covectors, you'll be left with a tensor that's still "hungry" for a few things.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Introduction GA_googleFillSlot("encyclopedia_square");

Visualizing one-forms

Basis of the dual space

Differential one-forms

See also

Reference

Introduction