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Topic: Contravariant

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	Tensors, Contravariant and Covariant
		Thus when we use orthogonal coordinates we are essentially using both contravariant and covariant coordinates, because in such a context the only difference between them (at any given point) is scale factors.
		Thus the metric of a polar coordinate system is diagonal, just as is the metric of a Cartesian coordinate system, and so the contravariant and covariant forms at any given point differ only by scale factors (although these scale factor may vary as a function of position).
		This is not ordinarily done, but it is possible. Recall that the contravariant components are measured parallel to the coordinate axes, and the covariant components are measured normal to all the other axes.
	www.mathpages.com /rr/s5-02/5-02.htm (2511 words)

	Pellionisz (1985) Tensor Network Theory of the Metaorganization of Functional Geometries in the Central Nervous System
		The contravariant character of the forces exerted by the muscles follows directly from the fact that in a steady-state the active forces must balance; the muscle components must yield a resultant force that is equal but opposite in direction to the load (G, see Fig.
		Thus a contravariant metric tensor is implied in the physical geometry of a motor apparatus.
		The co- and contravariant inter-relations evoked by external physical reality can manifest themselves in case of a lack of an explicit realization or even in case of a total absence of an ordinary metric: both sensory and motor processes are possible without an intermediate co-ordinated transfer.
	usa-siliconvalley.com /inst/pellionisz/85_metaorganization/85_metaorganization.html (9754 words)

	Contravariant and Covariant Vector Fields
		The tranformation rule for all contravariant vector fields is therefore given as follows.
		Question Geometrically, a contravariant vector is a vector that is tangent to the manifold.
		This gradient is, in general, neither covariant or contravariant.
	people.hofstra.edu /Stefan_Waner/diff_geom/Sec4.html (1501 words)

	PlanetMath: tensor product (vector spaces)
		These mappings were called “second-order contravariant tensors” and their values were customarily denoted by superscripts, a.k.a.
		Of course, it was understood that geometrically no one basis could be preferred to any other, and this leads directly to the definition of geometric entities as lists of measurements modulo the equivalence engendered by changing the basis.
		This relationship is the source of the terminology “contravariant tensor” and “contravariant index”, and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of “contravariant functor”;.
	planetmath.org /encyclopedia/TensorProduct2.html (599 words)

	PlanetMath: tensor array
		are multidimensional arrays with two types of (covariant and contravariant) indices.
		, it is customary to write contravariant indices using superscripts, and covariant indices using subscripts.
		We also mention that it is customary to use columns to represent contravariant index dimensions, and rows to represent the covariant index dimensions.
	planetmath.org /encyclopedia/TensorArray.html (486 words)

	Untitled
		In a limited context, the terms "Covariance" and "Contravariance" arise in cases where you have a basis for a vector space that is NOT orthonormal.
		In the restricted context of the presence of a metric, one way to view covariance and contravariance is to think of them as different basis descriptions of the same object.
		In the physical sciences "Contravariant" applies to vectors that arise from the derivative of the position vector, such as velocity and acceleration.
	home.pacbell.net /bbowen/covariant.htm (610 words)

	Pellionisz: Tensor Model of Gaze Control
		Let a physical invariant be represented by an i-dimensional contravariant vector, v' (with physical components), and let us calculate the covariant components of the same invariant along the j axes of the ul covariant vector (let i=5, and j=4, although there will be no restrictions to either i or j).
		This contravariant vectorial version of the physical invariant is amenable to the covariant sensorimotor embedding procedure, which yields a vectorial expression in the motor frame, but in projectiontype covariant components.
		In this sense, both the contravariant generation of the invariant and the covariant vectorial measurement of the same is available in the vestibulo-collicular sensorimotor reflex (cf., the scheme in Pellionisz (1984b) and in Chapter 15).
	usa-siliconvalley.com /inst/pellionisz/berthoz/berthoz.html (8695 words)

	The Universe of Discourse : Contravariant types
		If we adopt the rule of thumb that most values support few operations, and a few values support some additional operations, then the containment relation for functionality is contravariant to the containment relation for sets.
		I remember standing on a train platform around 1992 and realizing this for the first time, that containment of function types was covariant in the second component but contravariant in the first component.
		I suspect that the use of "covariant" and "contravariant" here suggests some connection with category theory, and with the notions of covariant and contravariant functors, but I don't know what the connection is.
	blog.plover.com /CS/contravariant.html (1052 words)

	PlanetPhysics: covariance and contravariance (Site not responding. Last check: 2007-09-30)
		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 r2dÎ¸ and its integral depends on the path.
		Contravariance is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds.
		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.
	planetphysics.org /encyclopedia/CovarianceAndContravariance.html (1326 words)

	Contravariant (Site not responding. Last check: 2007-09-30)
		Contravariant is a mathematical term with a precisedefinition 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 coordinatesystem being used as the basis of the tensor.
		Notice the superscript, this is a standard nomenclature convention for contravariant tensor components and should not beconfused with the subscript; which is used to designate covariant tensorcomponents.
	www.therfcc.org /contravariant-43943.html (366 words)

	Science Fair Projects - Contravariant
		In 2 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.
		In more modern terms, the transformation properties of the covariant indecies of a tensor are given by a pullback; by contrast, the transformation of the contravariant indecies is given by a pushforward.
		In category theory a functor may be covariant or contravariant, with the dual space being a standard example of a contravariant construction and tensor.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Contravariant (564 words)

	[No title]
		For example, the holonomy of a connection may be non-discrete when the connection is flat, contravariant connections cannot be pushed back or forward, etc. However, just like in ordinary geometry, contravariant connections are useful to study global properties of Poisson manifolds.
		Using generalized contravariant connections we show that we have a notion of \emph{Poisson holonomy} of the symplectic foliation, analogous to the holonomy in the theory of regular foliations.
		However, there is an appropriate notion of a basic connection on $M$: these are linear contravariant connections which preserve the Poisson tensor and restrict in each leaf to the Bott contravariant connection.
	www.intlpress.com /journals/JDG/archive/vol.54/issue2/2_3 (6247 words)

	tensor transformations - Science Forums (Site not responding. Last check: 2007-09-30)
		Now that you know the transformations for covariant and contravariant vectors, the same method can be used to determine the transformations for a tensor.
		The key point is realizing that a tensors of n covariant and m contravariant ranks maps n contravariant and m covariant vectors on a scalar.
		Contravariant vectors are elements of the dual space.
	www.scienceforums.net /forum/showthread.php?t=13793 (1242 words)

	Tensors, Contravariant and Covariant
		As can be seen, the jth component of the "contravariant path" from O to P consists of a segment parallel to jth coordinate axis, whereas the jth component of the "covariant path" consists of a segment perpendicular to all the axes other than the jth.
		In the special case where the determinant of the metric tensor is 1, the scale factor drops out and we can say that the contravariant and covariant versions of a vector are really both just ordinary contravariant representations of the same vector based on mutually dual coordinate systems.
		Recall that the contravariant components are measured parallel to the coordinate axes, and the covariant components are measured normal to all the other axes.
	www.physics.uq.edu.au /people/ross/phys2100/tensors.htm (3207 words)

	Maxima Manual: 27. itensor
		In this notation, contravariant indices are inserted in the appropriate positions in the covariant index list, but with a minus sign prepended.
		This is a function of 3 groups of indices which represent the covariant, contravariant and derivative indices.
		When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction.
	maxima.sourceforge.net /docs/manual/en/maxima_27.html (4776 words)

	[No title] (Site not responding. Last check: 2007-09-30)
		The chain rule for differentiation that you use to express dr/dt and dtheta/dt in terms of dx/dt and dy/dt says that dx/dt = dx/dr dr/dt + dx/dtheta dtheta/dt dy/dt = dy/dr dr/dt + dy/dtheta dtheta/dt This is precisely telling you that velocity transforms as a contravariant vector.
		That's just how it is. If you want to get a contravariant vector out of acceleration, you need to use the connection.
		You will find that a velocity is indeed a (tangent) contravariant vector, but its derivative is not.
	www.math.niu.edu /~rusin/known-math/99/acceleration (348 words)

	[No title]
		Contravariant indices are preceded by a hat, ^.
		- contravariant basis indices - indices that follow a semi-colon, `;', indicate covariant derivatives.
		grdef (`A{[a b] (c d)}`): - The covariant or contravariant nature of the braces are determined by the index which immediately follows an open brace (, [, or the index which immediately precedes a close brace), ].
	grtensor.phy.queensu.ca /Griihelp/grdef.help (1358 words)

	Sixth Chapter: Full Motor Coordinate VOR Model
		The goal of this chapter is to solve the computational problem of transforming the covariant vectors of head rotation in six dimensions to the contravariant vectors of eye motor commands, also in six dimensions, required to satisfy the VOR.
		The dual basis to the canal basis is introduced, as well as a method for estimating the contravariant coordinates of the eye position vector from its covariant coordinates.
		The problem of calculating the contravariant coordinates of the eye position vector in the overcomplete motor coordinate system was solved by Pellionisz using the Moore-Penrose generalized inverse.
	www.cnl.salk.edu /~olivier/olivier/intro_th/node7.html (894 words)

	covariant vs contravariant
		It seems that contravariant fields are just the normal vector fields they introduced in multivariable calculus, but if so, I can't figure out what covariant fields are.
		A covariant vector is specifically a vector which transforms with the basis vectors, a contravariant vector on the other hand is a vector that transforms against the basis vectors.
		A "contravariant vector" is what you'd simply call a "vector" which comes in two flavors.
	www.physicsforums.com /showthread.php?t=58257 (2642 words)

	CSDC : Cartan's Calculus: the interior product.
		However, a contravariant vector field on the final state is not well defined on the initial state if the inverse Jacobian matrix does not exist.
		Hence, the inner product of a p-form and a contravariant vector density pulled back is well defined in terms of the interior product of the pullback of the p-form and the pull back of the contravariant vector density.
		The interior product is often utilized in terms of a contravariant vectors, not contravariant vector densities, but then the result is only well defined with respect to diffeomorphisms, for which the inverse Jacobian is defined.
	www22.pair.com /csdc/ed3/ed3fre5.htm (460 words)

	covariant vs. contravariant
		The necessity of both "covariant" and "contravariant" objects is due to the existence of the canonical isomorphism (as vector space) between a vector space and its algebraic dual.
		The way I see it is that covariant vectors (note not covariant components) transform with the change in co-ordinates, and contravariant vectors transform against the the change in co-ordinates.
		Imagine a parallel plate capacitor with a displacement vector d between the plates and and an electric field vector E between the plates.
	www.physicsforums.com /showthread.php?threadid=148660 (893 words)

	User defined equalities and relations
		Morphisms can also be contravariant in one or more of their arguments.
		A morphism is contravariant on an argument associated to the relation instance R if it is covariant on the same argument when the inverse relation R
		Notice that division is covariant in its first argument and contravariant in its second argument.
	coq.inria.fr /V8.1beta/refman/Reference-Manual024.html (2406 words)

	[No title]
		Lemma 1.5 Let X be a contravariant C-space, Y be a covariant C-space and Z b* *e a space.
		In particular Lemma 1.5 says that for a fixed covariant C-space Y the fun* ctor - C Y from the category of contravariant* C-spaces to the category of spaces and the f* unctor map (Y; -) from the category of spaces to the category of contravariant* C-spac* *es are adjoint.
		Suppose that X is a contravariant functor from C to CW -COMPLEXES, i.* *e.
	hopf.math.purdue.edu /DavisJ-Lueck/assembly.txt (17837 words)

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