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Topic: Covariant differentiation

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	PlanetMath: connection
		In fact, in differential geometry, the definition of a curved space is a space in which there exist two distinct curves with the same endpoints such that parallel transport along one curve is not the same as parallel transport along the other curve.
		This generalization of differentiation involving parallel transport is known as covariant differentiation.
		Sometimes, as in the theory of embedded surfaces, there are two connections present so a semicolon is used to indicate covariant derivatives with repsect to one connection and a vertical bar or a colon is used to indicate covariant derivatives with respect to the other connection.
	planetmath.org /encyclopedia/ChristoffelSymbol.html (3013 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		At first, the theory of covariant differentiation was constructed on Riemannian manifolds and was intended in the first instance for the investigation of the invariants of differential forms.
		The definition and properties of covariant differentiation subsequently proved to be related in a natural way with the notions of connection and parallel displacement on manifolds, which were introduced later.
		The value of covariant differentiation is that it provides a convenient analytic apparatus for the study and description of the properties of geometric objects and operations in invariant form.
	eom.springer.de /c/c026870.htm (880 words)

	Covariant Differentiation (Site not responding. Last check: 2007-11-06)
		Covariant differentiation is accomplised by first taking the ordinary derivative of the object in question, and then forming contractions of each of the objects' indices with those of the Christoffel symbol:
		The printed output of a covariant derivative is similar to that for ordinary derivatives: the printname is that of the parent and the index is printed with the covariant derivative operator in place.
		Covariant differentiation is not restricted to tensors alone, objects with other types of indices may also be arguments to cov().
	www.scar.utoronto.ca /~harper/redten/node36.html (674 words)

	Cartan connection - Wikipedia, the free encyclopedia
		In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan.
		Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames.
		It is precisely the differential structure which is inherited from the differential structure of the Lie group which endows these homogeneous spaces with more structure (of a differential kind) than homogeneous spaces in general.
	en.wikipedia.org /wiki/Cartan_connection (1328 words)

	connection (mathematics) (Site not responding. Last check: 2007-11-06)
		In differential geometry, connection (spelt as connexion by the British) is a way of specifying covariant differentiation on a manifold.
		In one particular approach, a connection is a Lie algebra valued 1-form which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative.
		A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act on vector bundle sections.
	www.yourencyclopedia.net /Connection_(mathematics) (300 words)

	[No title]
		Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.
		The covariant indices are specified by a list as the first argument to the indexed object, and the contravariant indices by a list as the second argument.
		Differentiation of an indexed object with respect to some coordinate whose index does not appear as an argument to the indexed object would normally yield zero.
	www.unf.edu /public/cap4630/kmartin/gradfall94/maxima/tensor/manual.txt (4543 words)

	Covariant Differentiation
		The covariant derivative is stored on the input object's property list under the key cov, in the same format as used for ordinary derivatives; whenever cov() is called it looks there first to determine if the covariant derivative object already exists.
		The printed output of a covariant derivative is similar to that for ordinary derivatives: the printname is that of the parent and the index is printed with the covariant derivative operator in place.
		Covariant differentiation is not restricted to tensors alone, objects with other types of indices may also be arguments to cov().
	www.utsc.utoronto.ca /~harper/Redten/redten/node36.html (674 words)

	MovingSurfaces.html (Site not responding. Last check: 2007-11-06)
		The covariant basis is a (vector-valued) tensor which can be shown by a simple application of the chain rule.
		Covariant derivative reduces to the partial when applied to tensors of order zero, or when applied to general tensors in affine coordinate systems.
		Since the second term is zero, we obtain the formula for differentiating the shift tensor.
	www.math.drexel.edu /~pg/fb/docs/MovingSurfaces.html (643 words)

	Curvature of Riemannian manifolds
		Category:Riemannian geometry In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point.
		The articles Cartan connection and covariant derivative explain two different ways to introduce and calculate the curvature tensor.
		the curvature tensor measures anticommutativity of the covariant derivative.
	encyclopedia.codeboy.net /wikipedia/c/cu/curvature_of_riemannian_manifolds.html (892 words)

	covariant vs contravariant
		But the normal differentiation does not work because the coordinate lines are curved and normal partial derivitive destroys your vector, it won't be the vector anymore (you can check it youself).
		The difference between the normal differential and 'covariant differentiation' is defined by Christoffel symbols.
		Because 'covariant differentiation' was invented to connect vector in one point with vector in another point, it's called sometimes 'connection'.
	www.physicsforums.com /showthread.php?p=429688 (2475 words)

	Covariant Derivative (Site not responding. Last check: 2007-11-06)
		Covariant derivative expansion of the Yang-Mills effective action at high temper...
		A spacetime whose invariant classification requires the fifth covariant derivati...
		Covariant derivative expansion of fermionic effective action at high temperature...
	www.scienceoxygen.com /math/523.html (141 words)

	Curvature tensor
		In differential geometry, the curvature tensor is one of the most important notions; it generalizes Gauss curvature to higher dimensions.
		The infinitesimal geometry of Riemannian manifolds with dimension ≥ 3 is too complicated to be described by one number at a given point.
		All three of these give equivalent structure - parallel transport is equivalent to specifying a covariant way of differentiating - or a connection; and a connection determines the curvature tensor.
	www.guajara.com /wiki/en/wikipedia/c/cu/curvature_tensor.html (728 words)

	Covariant Differentiation (Site not responding. Last check: 2007-11-06)
		What he had really proven in [1; (3.129-131)] and [2; (J.18-20)] is a well-known identity, which guarantees the independency of the total differential of a vector field of the used reference frame (compatibility identity).
		This is extended to covariant differentiation of covectors (1-forms) by means of the somewhat deeper concept of covectors (cf.
		The analogue of (1.14) for the covariant derivatives is
	www.mathematik.tu-darmstadt.de /~bruhn/covar_deriv.htm (2405 words)

	Computational and Applied Mathematics Group (CAM)
		Ck-manifolds, charts, an atlas of charts, transformation properties of scalar fields and differentials; multilinear forms; intrinsic and axiomatic definitions of (contravariant) vectors and (covariant) covectors (at a point on the manifold); intrinsic and axiomatic definitions of general rank (r,s) tensors; vector, covector, and general rank (r,s) tensor fields over a manifold.
		Covariant differentiation; affine connections; covariant differention of vectors, covectors, and general rank (r,s) tensors; Tangent and co-tangent spaces, tangent and co-tangent bundles, fibers of tangent and co-tangent bundles; Natural projection and its inverse; covariant differential.
		Non-commutativity of covariant differentiation: Riemann and torsion tensors; Flatness of the metric tensor and vanishing Riemann tensor.
	cam.ucsd.edu /~mholst/teaching/ucsd/237a_w07 (703 words)

	Re: Covariant vs Absolute Derivative
		He is calling >> P;_(whatever) a covariant derivative which is fine as it is, but what I am >> saying is that it is the partial derivative aspect of covariant >differentiation >> and calling it just by covariant derivative does not make the DP/dtau >operation >> a non frame covariant operation.
		If you only want to call a covariant >partial >> derivative by covariant derivative I don't care so much, my main point was >that >> when a function is only a function of the one independent variable then >the >> partial derivative with respect to that variable becomes just a strait >> derivative.
		This is the case in ordinary differentiation just as much as >it is >> the case in frame covariant differentiation.
	www.usenet.com /newsgroups/sci.physics.relativity/msg01659.html (476 words)

	CSDC : Why Topology?
		A more general definition of "Covariant" differentiation is based upon the notion of a "Connection", and does not depended upon the existence of a metrical distance.
		The idea of a connection is to define a differential process such that when it operates on a tensor, the resultant object is also a tensor.
		Cartan's theory of exterior differential forms can give part of the answer, for it appears that Cartan's methods can be applied to problems of continuous topological evolution.
	www22.pair.com /csdc/ed3/ed3fre2.htm (775 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Differentiation of Vector Fields along Curves in Rn The Geometry of Space Curves 292 Curvature of Plane Curves 296 2.
		Differentiation of Vector Fields on Submanifolds of Rn Formulas for Covariant Derivatives 303 Differentiation of Vector Fields 305 3.
		Addenda to the Theory of Differentiation on a Manifold The Curvature Tensor 316 The Riemannian Connection and Exterior Differential Forms 5.
	www.math.harvard.edu /graduate/books/boothby.html (416 words)

	Maxima Manual: 27. itensor
		Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices.
		The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).
		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)

	Theory Continued page 3
		Second covariant derivatives generally are not independent of order, and their commutated value depends linearly on the original tensor, and the coefficients form a matrix-valued 2-form called the Riemann-Christoffel tensor.
		After differentiation and subtraction it is seen that the value is a matrix-valued 2-form.
		The covariant derivative of the vector field measures the vectors' deviation from parallelism, and is zero at x2 when the field is parallel at x2.
	www.superstringtheory.fanspace.com /custom4.html (3489 words)

	[No title]
		Moreover the passage from one coordinate system to the another is smooth in the overlapping regions, so that the meaning of differentiable curve, function, or map is consistent when referred to either system.
		Thus the velocity V in terms of the basis vectors using the dot notation for time differentiation is:
		This differential equation defines the kth coordinate of the path taken by a particle not under the influence of any external forces in the geometry defined by the coordinate systems of the manifold.
	www.realtime.net /~welbon/Christoffel_Symbols.html (439 words)

	Einstein Relativity Page 152 (Site not responding. Last check: 2007-11-06)
		We shall now introduce the covariant exterior derivative.
		The covariant exterior derivative of the form-valued vector-components are
		The covariant exterior derivative of these components is defined by
	www.chemical-changes.com /Einstein.relativity/Einstein.relativity-152.html (85 words)

	Module and Programme Catalogue
		It also predicts various exotic phenomena like fl holes, and provides the basis for possible cosmological models of the life history of the universe, usually starting with a 'big bang'.
		This module should make a good companion to options on differential geometry, although no previous knowledge of the latter will be assumed.
		Covariant derivative; curvature tensor; Bianchi identity, Ricci and Einstein tensors, Einstein's field equations; surfaces and Gaussian curvature; spaces of constant curvature.
	www.leeds.ac.uk /modules/200203/ug/math3443.htm (329 words)

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