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Topic: Covector

	PlanetMath: covector
		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.
	planetmath.org /encyclopedia/Covector.html (73 words)

	Math Forum - Ask Dr. Math
		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.
	mathforum.org /library/drmath/view/51503.html (639 words)

	Cotangent space - Wikipedia, the free encyclopedia
		That is, given a tangent covector there is no canonical tangent vector associated with it.
		M and hence it is a tangent covector at p.
		If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.
	en.wikipedia.org /wiki/Cotangent_space (793 words)

	what do the columns and rows of the metric correspond to? (Site not responding. Last check: 2007-10-15)
		A covector is merely something you can "dot into" a vector to yield a real number (the dot product).
		By definition, covectors are "dual" to vectors (and vice-versa).
		At base the metric is a tensor that establishes an isomorphism between vectors and covectors.
	www.pych-one.com /new-5663135-4395.html (1704 words)

	1-Forms
		At a particular point on the manifold, the local linear approximation to the contours is a "covector" (or dual vector) and it's represented as a collection of hyperplanes in the tangent space at that point.
		And the covector associated with a 1-form at each point is the same, whether or not the 1-form globally represents a gradient.
		The "basis covector" corresponding to a particular "basis vector" is the covector which points in the same direction as that basis vector, and which has a wavelength exactly as long as the basis vector.
	www.physicsinsights.org /pbp_one_forms.html (1120 words)

	Re: The Hodge dual: some definitions & examples
		>Covectors are >'linear functionals', so things like w(v) make sense and are scalars >when w is a covector and v is a vector.
		2) Vectors and covectors (and presumably their higher order equivalents) are objects, and the basis used is irrelevant to how they are manipulated.
		The covector defines 'spacing between planes', that is it in some sense defines the effect generated by the vector.
	www.lns.cornell.edu /spr/2004-11/msg0065213.html (1728 words)

	Glossary
		Any linear map of covectors to the real numbers is called contravariant, and any linear map of vectors to the real numbers is called covariant.
		Covector -- also called a dual vector -- A linear map from vectors into the real numbers.
		A covector maps vectors into real numbers and is hence a "rank (0,1)" tensor.
	www.physicsinsights.org /glossary.html (1220 words)

	7
		An alternative interpretation is that the metric is a map which relates vectors in a vector space to their images in a dual, or covector, space.
		In this picture there are two distinct vector spaces, and every vector in the vector space has exactly one associated partner in the dual space (and vice versa).
		We discussed how the role played by vectors and covectors is analogous to that played by column vectors and row vectors in flat space.
	www.emory.edu /PHYSICS/Faculty/Benson/380-04/notes/7/7.html (817 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		This associates with x^\mu, a vector, a different kind of object, called a covector.
		We saw that the effect on components of lowering x^\mu to x_\mu, by matrix multiplying by g_{\mu\nu}, is just to multiply all the spatial components by a minus sign.
		To get the magnitude of a four-vector, which we represent as a column vector, we must multiply it by its covector --- however that covector is not just its transpose, but the matrix g multiplied times the four-vector, then transposed.
	www.physics.emory.edu /faculty/Benson/380-96/notes/6.txt (770 words)

	Tensor Vectorfield probe Visualization (Site not responding. Last check: 2007-10-15)
		Since the covector field is orthogonal to the vector field (the cotangential space is dual to the tangential space), it may be visualized using planes:
		Less known is the appearance of the coordinate vector field and the coordinate covector field of the polar theta-coordinate.
		Contraction of a covector and a vector gives a number, as does a tensor of rank two when two vectors are provided.
	www.zib.de /benger/TensorViz (307 words)

	Covariant derivative - Wikipedia, the free encyclopedia
		Given a field of covectors (or one-form) α, its covariant derivative
		The covariant derivative of a covector field along a vector field v is again a covector field.
		Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields using the following identities where
	en.wikipedia.org /wiki/Covariant_derivative (1163 words)

	Why is 4-momentum usually given as a covector? Text - Physics Forums Library
		10-07-2006, 07:25 AM I understand the association between vectors and covectors on a Riemannian manifold, but it appears that 4-momentum is given naturally as a covector, instead of vector.
		Momentum as a scalar conserved quantity is naturally a covector.
		10-07-2006, 12:51 PM I understand the association between vectors and covectors on a Riemannian manifold, but it appears that 4-momentum is given naturally as a covector, instead of vector.
	www.physicsforums.com /archive/index.php/t-135193.html (1467 words)

	Re: The Hodge dual: some definitions & examples
		So if you are given a covector L, there is a vector w so that L=*w.
		Next, I want to point out that the collection covectors has the property that things can be added and multiplied by scalars, so the collection of covectors is a vector space in its own right.
		But the vector and covector are the things related and will be the geometrical things when a manifold is in the picture.
	www.lns.cornell.edu /spr/2004-11/msg0065292.html (1403 words)

	Untitled
		In actuality though, vectors and covectors are more generally defined than in the above conceptual model, but in the case of the existence of a metric, the model is equivalent and will take you a long way.
		Vectors and Covectors, Contravariance and Covariance, are generally defined independent of a metric.
		The general definition of a covector is, "a linear real valued function of a vector" or a function that takes in a vector and outputs a real number.
	home.pacbell.net /bbowen/covariant.htm (610 words)

	How to show something is a covector? Text - Physics Forums Library
		(df/dx, df/dy,df/dz,df/dt) (f is a scalar field) is a covector
		When you have a covector and a vector, you're just plugging the vector into the covector.
		(Since a covector is a linear functional) In the coordinate representation, it's multiplying a 1xN matrix by an Nx1 matrix.
	www.physicsforums.com /archive/index.php/t-114019.html (365 words)

	Training Analysis - Super Soaker Central
		Covector was the loner, while Quantum, Cobra, and I looked for him.
		The codeword was yelled and I came running to help Quantum, unfortunately, Quantum was taken out before she could hit Covector.
		Covector, with one of two hits, and Cobra, with no hits, where the only remaining, I took this opportunity to drop my weapon and begin to take pictures from all angles & view points.
	www.sscentral.org /stories/trainan.php (556 words)

	8
		We reviewed the maps between four-vector and covector introduced last time, and showed that taking a vector to its covector and back reproduced the original vector.
		We showed that under any spacetime coordinate transformation (of which Lorentz transformations are a subgroup), this operator must transform as a covector, because the coordinates remain independent.
		We thus now have all the elements to build Lorentz-covariant laws of physics -- grouping observables and their derivatives into 4-vectors, covectors, and tensors with known Lorentz transformation properties; then generalizing equations so that both left and right sides transform in definite, and equivalent, ways under Lorentz transformations.
	www.emory.edu /PHYSICS/Faculty/Benson/380-04/notes/8/8.html (268 words)

	outline1.html
		A COTANGENT VECTOR or simply COVECTOR at the point p is a function f that eats a tangent vector v and spits out a real number f(v) in a linear way.
		This is obvious for the rank (0,k) tensors, but for the rank (1,k) ones we need to check that a function that eats k vectors and spits out a vector v can be reinterpreted as a function that eats k vectors and one covector f and spits out a number.
		Similarly, note that a vector can be reinterpreted as a tensor of rank (1,0), and a covector can be reinterpreted as a tensor of rank (0,1).
	math.ucr.edu /home/baez/gr/outline2.html (3217 words)

	[No title]
		\endproof \end A {\bf covector} $\omega$ is a real valued function on ${\bf R}^n$, so $$ \omega:{\bf R}^n\rightarrow{\bf R},$$ having the following two properties: \vs (1) $\omega(c\b{x})\,=\,c\omega(\b{x})$ for any scalar $c$ and and vector $\b{x}$ and \vs (2) $\omega(\b{x}+\b{y})\,=\,\omega(\b{x})+\omega(\b{y})$ for any vectors $\b{x}$ and $\b{y}$.
		Then check that covectors behave the same way as vectors with respect to these operations.
		For each $i=1,\ldots,n$ we let $$\b{e}^i$$ be the covector whose value on the vector $\b{x}$ is the $i$-th component of $\b{x}$.
	www.math.duke.edu /~wka/math103/junk (589 words)

	HDF5 Chart Objects
		A chart object is just a named reference point (like an anchor), which is used to allow detection of compatibility among various data groups.
		It also defines named types for Points, Vectors, Covectors, Metrices and similar data types.
		In general, covariant (lower) indices are denoted via a lower case `d', kontravariant (upper) indices via an upper case `D' preceding the coordinate name.
	www.zib.de /benger/F5/Charts.html (583 words)

	PlanetMath: tensor
		To establish (2), we note that the transformation rule for the dual basis takes the form
		The transformation rule for covector components is covariant.
		In light of the above discussion, we see that the transformation rule for a general type
	planetmath.org /encyclopedia/Tensor.html (850 words)

	Problem 2.1 (Site not responding. Last check: 2007-10-15)
		The representation of this tensor, which maps pairs of covectors to numbers, follows the representation of covectors, where we use the set of vectors which are mapped to one, and the representation of metric tensors, where we use the set of vectors which are mapped to one.
		We use the set of covectors which are mapped to one by the tensor, and plot them as lines in the vector space.
		If we write the covector (finally we understand this notation) (careful, these L's may look like 1's on some screens)
	www.ucolick.org /~burke/class/prob21.html (260 words)

	mat531 week6
		A covector at a point x in M is a linear map omega : TM_x --> R. The set of such linear maps is the vector space T*M_x, the co-tangent space at x.
		For example, if f is a smooth function defined near x, then f defines the covector df_x(v) = v(f).
		These covectors form a basis for the cotangent space: if omega(D/Dx^a_i) = A^i, then omega = sum_i A^i dx^a_i.
	www.math.sunysb.edu /~tony/archive/top2/week6.html (621 words)

	Constant Covectors (Site not responding. Last check: 2007-10-15)
		Push a constant covector onto the execution stack.
		Before being pushed onto the execution stack, the covector is transformed by the inverse transpose of the current modelview matrix.
		Use a Normal if you want an automatically reparameterized object that is transformed like a covector.
	www.cgl.uwaterloo.ca /Projects/rendering/SMASH/spec_0.2/ConstantCovectors.html (109 words)

	Oriented Matroids
		A non-zero covector with inclusion-minimal support is called a cocircuit.
		are of the same rank, and for every covector
		The reader may get a picture of covectors by thinking of some linear oriented hyperplane (corresponding to the
	www.uni-bayreuth.de /departments/wirtschaftsmathematik/rambau/Diss/diss_MASTER/node37.html (202 words)

	On the braided Fourier transform in the n-dimensional quantum space (ResearchIndex)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: : We work out in detail a theory of integrability on the braided covector Hopf algebra and the braided vector Hopf algebra of type An introduced in [Ma] and [KeMa].
		Starting with a definition of braided Fourier transform very similar to that in [KeMa] we obtain n-dimensional analogous results to those in [Koo] expressing the correspondence between products of the q 2 -Gaussian g q 2(x) times monomials, and products of the q 2 -Gaussian G q 2(@) times q 2 -Hermite polynomials under the...
	citeseer.ist.psu.edu /159794.html (487 words)

	Oriented Matroids - References and Bibliography (Site not responding. Last check: 2007-10-15)
		An oriented matroid can be represented (and defined) in several, equivalent ways.
		The basic axiom systems include vector (or covector) axioms, circuit (or cocircuit) axioms, and chirotopes.
		We define here an oriented matroid using the covector axioms: An oriented matroid is a pair (E, F) of a finite set E and a set F of sign vectors (called covectors) on E for which the following covector axioms (F0) to (F3) are valid:
	www.om.math.ethz.ch /?p=bib&glo=om (120 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		The dimensions of a vector, covector and matrix are set at the Values menu (see below).
		This menu lets you select the dimension of vectors, covectors and matrices.
		The presentation of bool, integer and real values can be set to colors, values (meaning: numbers) and arrows.
	lienhard.desy.de /mackag/cs/cl/syclhome/sycldoc.html (638 words)

	Nonlinear connections and spinor geometry (Site not responding. Last check: 2007-10-15)
		We present an introduction to the geometry of higher-order vector and covector bundles (including higher-order generalizations of the Finsler geometry and Kaluza-Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures.
		We emphasize strong arguments for application of Finsler-like geometries in modern string and gravity theory, noncommutative geometry and noncommutative field theory, and gravity.
		Manuscript submissions are open for the new special issue on Composite Hollow Spheres and Capsules
	www.hindawi.com /GetArticle.aspx?doi=10.1155/S0161171204212170 (101 words)

	Oriented Matroid Pairs, Theory and an Electric Application (ResearchIndex)
		The property that a pair of oriented matroidsM ?
		L, MR on E have free union and no common (non-zero) covector generalizes oriented matroid duality.
		This property characterizes when certain systems of equations whose only nonlinearities occur as real monotone bijections have a unique solution for all values of additive parameters.
	citeseer.ist.psu.edu /97919.html (695 words)

	Amazon.com: "tensor gab": Key Phrase page (Site not responding. Last check: 2007-10-15)
		Now it turns out that from rbe one can build up a special tensor Gab that automatically has vanishing divergence.
		Key Phrases in this book: Properties of Riemann, contravariant vector field, fundamental observers, metric gab, geodesic equation, covariant vector field, tensor gab, null geodesics, vacuum equations, geodesic deviation, covector fields, affine parameter (See more)
		Key Phrases in this book: Properties of Riemann, contravariant vector field, fundamental observers, metric gab, geodesic equation, covariant vector field, tensor gab, null geodesics, vacuum equations, geodesic deviation, covector fields, affine parameter, vanishing divergence, isometry group, future light cone, coordinate patch, transition law, metric connection, covariant differentiation, inhomogeneous term (
	amazon.com /phrase/tensor-gab (571 words)

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