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Topic: Covariant transformation

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	Covariant derivative - Wikipedia, the free encyclopedia
		Here we give a traditional index-notation introduction to the covariant derivative (also known as the tensor derivative) of a vector with respect to a vector field; the covariant derivative of a tensor is an extension of the same concept.
		The covariant derivative can be described by a tensor in a fixed coordinate chart, but it is not a tensor in the sense that it is not invariant under coordinate changes.
		Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma.
	en.wikipedia.org /wiki/Covariant_derivative (1233 words)

	Covariant transformation - Wikipedia, the free encyclopedia
		The transformation that describes the new basis vectors in terms of the old basis, is defined as a covariant transformation.
		Entities that transform covariant (like basis vectors) and the ones that transform contravariant (like components of a vector and differential forms) are "almost the same" and yet they are different.
		It is "almost the same space",except that the elements of the dual space (called dual vectors) transform contravariant and the elements of the tangent vector space transform covariant.
	en.wikipedia.org /wiki/Covariant_transformation (1461 words)

	Pellionisz (1985) Tensor Network Theory of the Metaorganization of Functional Geometries in the Central Nervous System
		If covariant proprioception is used as a recurrent signal to the motor apparatus, as if it were a contravariant motor expression, then reverberations at their steady state yield the eigenvectors and eigenvalues of the system.
		The covariants are the passive force-components, measured as the orthogonal projections of the load-vector (G) onto the co-ordinate axes.
		Such transformation is determined by the principal direction-axes of the ellipsoid (given by the eigenvectors) and by the magnitude-distortion (where the lengths of the principal axes along each eigenvector correspond to the eigenvalue).
	usa-siliconvalley.com /inst/pellionisz/85_metaorganization/85_metaorganization.html (9754 words)

	Pellionisz, A. (1984) Coordination. (Cerebellar function with coordinates)
		An external invariant and its covariant and contravariant representation in the CNS.
		For sensory systems, the covariant feature is most obvious in the vestibular semicircular canal apparatus, where it is a physical fact that each individual canal responds to the orthogonal projection (cosine component) of the acceleration to the plane of the canal, and this action of one canal is independent of the action of the others.
		Cerebellar motor coordination: transformation of covariant intention to contravariant execution by the neuronal network acting as the Moore-Penrose generalized inverse of the covariant metric tensor of the motor space.
	usa-siliconvalley.com /inst/pellionisz/84_coordination/84_coordination.html (6388 words)

	Mathematics: N-Dimensional Numbers
		The determinant of the first matrix we denote as d (called the Jacobian of direct transformation) is inverse to the determinant of the second matrix we denote D=1/d (called the Jacobian of inverse transformation).
		Checking the transformation law shows us that it is not a second (third) rank tensor (we can not get these derivatives in transformed coordinates by applying the transformation law of a second (third) rank tensor).
		A one-point transformation of coordinate system is essentially an "arbitrary" transformation, because its main purpose is to introduce scalars, vectors, and tensors and separate mathematical objects from mathematical accessory.
	www.wbabin.net /yuri/keilman8.htm (5206 words)

	9.4.1 Some rules of tensor analysis on manifolds
		Covariant and contravariant: A lower label is often termed ``covariant'' and an upper label ``contravariant.'' The mnemonic ``co-low'' assists in remembering the terminology.
		Covariant and contravariant tensors can be considered dual, where the connection is through the metric tensor.
		Covariant derivative: In order to account for nonconstant unit vectors on a curved manifold, it is necessary to generalize partial derivatives to so-called covariant derivatives.
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node107.html (738 words)

	The Hole Argument
		The hole transformation that relates the two metric fields of the hole argument is an example of a gauge transformation (if we regard the two as physically equivalent).
		The decision as to whether a transformation is a gauge transformation cannot merely be decided by the mathematics; it is a physical issue that must be settled by physical considerations.
		A hole transformation is credibly a gauge transformation since that assumption protects the theory from an apparently illicit form of indeterminism.
	plato.stanford.edu /entries/spacetime-holearg (5340 words)

	Introduction to Tensors (Site not responding. Last check: 2007-10-09)
		The reason for the placement of the index on the quantity on the left side is due to the fact that this new quantity has the transformation properties of a covariant vector.
		In special relativity the transformations from one set of Minkowski coordinates to another set of Minkowski coordinates are referred to as Lorentz transformations and are defined as follows: let S and S’ be two inertial frames of reference where S’ is moving relative to S in the +x direction.
		This transformation maps the origin of S and the origin of S’ to (0, 0, 0, 0).
	www.geocities.com /physics_world/ma/intro_tensor.htm (1500 words)

	III. TRANSFORMATION RELATIONS
		The transformation relations from cartesian coordinates to a general curvilinear system are developed here using certain concepts from differential geometry and tensor analysis, which are introduced only as needed.
		The relation between the covariant and contravariant metric tensor components is obtained by use of Eq.
		With the time derivatives transformed, only time derivatives at fixed points in the transformed space will appear in the equations and, therefore, all computation can be done on the fixed uniform grid in the transformed field without interpolation, even though the grid points are in motion in the physical space.
	www.erc.msstate.edu /publications/gridbook/chap03/text.html (2615 words)

	Covariance - Psychology Central (Site not responding. Last check: 2007-10-09)
		This article is not about the physics topic, covariant transformation, nor about the mathematics example for groupoids, covariance in special relativity, nor about parameter covariance in object-oriented programming.
		That is to say, the covariance becomes more positive for each pair of values which differ from their mean in the same direction, and becomes more negative with each pair of values which differ from their mean in opposite directions.
		By contrast, the correlation, which depends on the covariance, is a dimensionless measure of linear dependence.
	psychcentral.com /psypsych/Covariance (454 words)

	No Title
		We further discussed the transformations of observables and operators under Lorentz boosts, en route to writing Maxwell's equations in a Lorentz covariant way.
		Redefining space and time (as Lorentz transformations do) induces redefinitions of other observables: for example, the velocity depends explicitly on space and time, and changes when those change due to a Lorentz transformation.
		Now that we have grouped the relevant observables and operators into objects that transform in a well-defined way under Lorentz tranformations (as vectors or covectors), we seek to write the laws of electromagnetism in a Lorentz-covariant way.
	www.physics.emory.edu /faculty/benson/380-96/notes/7/7.html (450 words)

	Tensors, Contravariant and Covariant
		Notice that each component of the new metric array is a linear combination of the old metric components, and the coefficients are the partials of the old coordinates with respect to the new.
		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.
		The transformation rule for such representations is more complicated than either (6) or (8), but each component can be resolved into sub-components that are either purely contravariant or purely covariant, so these two extreme cases suffice to express all transformation characteristics of tensors.
	www.physics.uq.edu.au /people/ross/phys2100/tensors.htm (3207 words)

	PHY423B CORE LESSONS & OUTPUT SKILLS, Summer '99 (Site not responding. Last check: 2007-10-09)
		State the Lorentz transformation for a boost along any coordinate axis in two forms (Lorentz factor γ, rapidity φ) and the relationship between the forms.
		Use their transformation properties to establish the transformation properties of the corresponding three-vectors.
		Use the transformation character of the electromagnetic field tensor to determine the transformation rules for the electric and magnetic fields in special cases.
	physnet2.pa.msu.edu /home/courses/423B/423Bcorelessons.html (1326 words)

	Tensors
		, it is assumed that the transformation in non-singular.
		However, since they transform like their differentials under linear homogeneous coordinate transformations, they do behave as tensors under such transformations.
		However, under linear transformations it behaves as a tensor, so under linear transformations the derivative of a tensor with respect to distance behaves as a tensor of the same type.
	farside.ph.utexas.edu /~rfitzp/teaching/jk1/lectures/node10.html (1038 words)

	Tensors as multilinear forms\\ Handout ``Methoden der Theoretischen Physik-\"Ubungen''
		To distinguish elements of the two bases, the covariant vectors are denoted by subscripts, whereas the contravariant vectors are denoted by superscripts.
		The entire argument concerning transformations of covariant tensors and components can be carried through to the contravariant case.
		Alexandrov's theorem states that the mere requirement of the preservation of zero distance (i.e., lightcones), combined with bijectivity of the transformation law yields the Lorentz transformations ([4,5,6,7,8] are original articles reviewed in [9,10]; see also [11] for an elementary proof).
	tph.tuwien.ac.at /~svozil/publ/2001-mu-tensor.htm (1480 words)

	General Relativity (Site not responding. Last check: 2007-10-09)
		In order to construct sensible theories one has to cast the physical laws in a manner that transformation properties are taken care of, so that laws retain their validity and observables their values irrespective of arbitrary choices of coordinate systems.
		In order to bring this in the canonical form for the transformation of a tensor, we have to get rid of the second term in the right-hand side of the equation, which arises from the derivative of the transformation coefficients.
		The curvature tensor is applied in the Einstein equation of the gravitational field, and is used to describe the motion of particles in curved spacetime.
	www.nikhef.nl /~henkjan/astro/node13.html (1621 words)

	Tensors, Contravariant and Covariant
		The gradient g = Ñy is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Suppose we have another system of smooth continuous coordinates X
		This is very similar to the previous formula, except that the partial derivatives are of the new coordinates with respect to the old. Arrays whose components transform according to this rule are called contra-variant tensors.
		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. This is the essential distinction (up to scale factors) between the contravariant and covariant ways of expressing a vector or, more generally, a tensor.
	mathpages.com /rr/s5-02/5-02.htm (1911 words)

	Einstein-Heuristics (Site not responding. Last check: 2007-10-09)
		Also, Einstein promoted the use of a formal mathematical principle, covariance, as a natural way to define what one meant by a "(general) law of physics," and then used it gain a considerable unification of physical theories.
		Having thus elevated covariance to a fundamental heuristic of his "formal point of view," it was a quick succession of steps to the unification of Galileo-Newtonian mechanics and Maxwell's optics.
		Thus, either Lorentzian covariance subsumes Galilean, or vice versa, or they are both related to each other by being subcovariances of some third covariance.
	www.ajnpx.com /html/Einstein-analects/Einstein-Heuristics.html (1256 words)

	Duke Physics: Duke PHY 318 Course Plan
		The main purpose is teach students the language and major concepts of electromagnetic theory in forms that are useful for physics research as well as advanced coursework in condensed matter physics, quantum field theory, optical physics, and nuclear physics.
		A minimal prerequisite for PHY 318/319 includes a good general physics course in electricity and magnetism and knowledge of mathematical techniques at the level of PHY 230; also, it is strongly recommended that students should have taken at least one semester of an intermediate undergraduate electromagnetism course at the level of Griffiths.
		With the possible exception of covariant electrodynamics, the discussion is highly readable and provides a good prelude to related material in Jackson.
	www.phy.duke.edu /graduate/courses/318plan.ptml (844 words)

	New Transformation Equations and the Electric Field Four-vector
		The Lorentz transformation equations are replaced by a new set of transformation equations, the electric field is described by a four-vector, and the analog of Maxwell's electromagnetic field tensor contains nonzero terms along the main diagonal, causing any relation that contains its components to include extra terms.
		Since these equations are used to determine the covariance of the laws of physics, any change in their form requires a change in the form and scope of these laws.
		Where previously it was unnecessary to distinguish between covariant and contravariant components of tensors, since they are equivalent in rectangular coordinates, we must now specify which we are using.
	www.softcom.net /users/der555/node22.html (3756 words)

	Quantum Mechanics - PhysicsWiki (Site not responding. Last check: 2007-10-09)
		One of the oldest and most commonly used formulations is the transformation theory invented by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).
		For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.
		Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation.
	wiki.advancedphysics.org /index.php/Quantum_Mechanics (4075 words)

	Background Lorentz Transformation (Site not responding. Last check: 2007-10-09)
		a background Lorentz transformation is a transformation of the form
		A covariant tensor of rank 2 transforms as
		transforms like a tensor under a background Lorentz transformation.
	www.geocities.com /physics_world/gr/back_lorentz_trans.htm (89 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Homework 2 We further discussed the transformations of observables and operators under Lorentz boosts, en route to writing Maxwell's equations in a Lorentz covariant way.
		Reviewed how the four-vector x^\mu = (ct, \vec{x}) transforms as a vector under Lorentz transformations: that is, the components ct and \vec{x} mix with each other in such a way that the four-vector x^\mu is just matrix-multiplied by a Lorentz tranformation matrix \Lambda.
		This object is a scalar: since it has 1 upper and 1 lower index, it is multiplied by one factor of \Lambda and one factor of \Lambda^{-1} under Lorentz transformations, which cancel to leave it invariant.
	www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/7.txt (711 words)

	The End of My Latin
		By the end of the 19th century the phenomena of electromagnetism had become well-enough developed so that the behavior of the electromagnetic field - at least on a macroscopic level - could be described by a set of succinct equations, analogous to Newton's laws of motion for material objects.
		At about this time, Lorentz derived the fact that although Maxwell's equations (taking the permissivity and permeability of the vacuum to be invariants) of the electromagnetic field are not covariant with respect to (1), they are covariant with respect to a complete set of velocity transformations, namely, those of the form
		Lorentz's equation [300] is simply the transformation law for electromagnetic forces, and his equations [305] give the relativistic expressions for the transverse and longitudinal masses of a particle. Lorentz has previously presented these expressions as
	www.mathpages.com /rr/s3-06/3-06.htm (418 words)

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