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Topic: Covariant derivative

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	Covariant derivative - Wikipedia, the free encyclopedia
		In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
		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/Tensor_derivative (1228 words)

	PlanetMath: connection
		Since the notions of connection, parallel transport, and covariant derivative are so closely related, it is easy to translate propositions involving one of these terms into propsitions involving a different one of three terms.
		This property is the origin of the term ``covariant derivative'' -- the covariant derivative maps tensor fields into quantities which transform in the same manner.
		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 (2998 words)

	covariant derivative (Site not responding. Last check: 2007-11-05)
		There is no real difference between the covariant derivative, and the connection concept — except for style in which they are introduced.
		Ocasionally covariant derivative refer to derivative of sectionss of a general vector bundle along tangent vector of the base, see subsection "Vector bundles" in "Connection form.
		The covariant derivative of a covector field along a vector field v is agin a a covector field.
	www.yourencyclopedia.net /Covariant_derivative.html (1189 words)

	Derivative (Site not responding. Last check: 2007-11-05)
		The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes.
		Derivatives are defined by taking the limit of the slope of secant lines as they approach a tangent line.
		If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum.
	hallencyclopedia.com /Derivative (2403 words)

	Exterior derivative - Wikipedia, the free encyclopedia
		In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree.
		The exterior derivative of a differential form of degree k is a differential form of degree k + 1.
		and the Lie derivative of a general differential form is closely related to the exterior derivative.
	en.wikipedia.org /wiki/Exterior_derivative (365 words)

	Covariant Derivative (Site not responding. Last check: 2007-11-05)
		It's much easier to "visualize" the covariant derivative using a higher dimensional Euclidean "scaffolding" into which you isometrically imbed your manifold (always possible, at least for spaces with a positive definite metric, but in anycase, the results are the same).
		It is the component of the everyday ol' Euclidean derivative that resides in the tangent space of the manifold.
		is the indexed vector field and ";j " and ";k " are the covariant derivatives of v with respect to parameters indexed by j and k, and repeated indices are summed.
	home.pacbell.net /bbowen/cov_deri.htm (275 words)

	2.1 Geometry (Site not responding. Last check: 2007-11-05)
		The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
		The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf.
		The covariant derivative of a tangent vector with bein-components
	relativity.livingreviews.org /Articles/lrr-2004-2/articlesu3.html (2680 words)

	Covariant Derivative (Site not responding. Last check: 2007-11-05)
		I know that with the affine connection the covariant derivative is a derivation because it is _defined_ that way.
		The curvature formed by the commutator of covariant derivatives will then have terms involving the holonomic partial derivative of the vielbein.
		If the covariant derivative is in fact a derivation then the extra terms become the covariant derivative acting on the veilbein rather than simply the partial - a big difference.
	www.lns.cornell.edu /spr/2002-03/msg0040419.html (245 words)

	The covariant derivative (Site not responding. Last check: 2007-11-05)
		A covariant derivative is a tensor which reduces to a partial derivative of a vector field in Cartesian coordinates.
		The covariant derivative of a contravariant tensor of rank one is given by
		The covariant derivative of a covariant tensor of rank one is given by the expression:
	ta.twi.tudelft.nl /isnas/isnas_mathmanual/node5.html (127 words)

	Re: Covariant Derivative (Site not responding. Last check: 2007-11-05)
		Covariant derivative (defined simply by parallel transport) is not a derivation.
		Covariant exterior derivative on the other hand leads from "p-form with values" to "p+1-form with values" and is a derivation provided exterior product of forms with values in an associated vector is defined (see Greub et al.,, Connections,...
		Covariant exterior derivative is always a derivation with respect to taking exterior products of vector valued forms with scalar valued forms.
	www.lns.cornell.edu /spr/2002-04/msg0040864.html (202 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)

	[No title] (Site not responding. Last check: 2007-11-05)
		The notion of a derivative is more complicated in a curved manifold than in the common case of flat geometry and Cartesian coordinates because the basis vectors will in general vary from point to point in the manifold.
		It is therefore no longer possible to identify the derivative of a tensor with the derivative of its components.
		In terms of a covariant derivative these terms are represented by the connection.
	gravity.psu.edu /~sperhake/Research/GRBasics/GRbasics.html (470 words)

	Outline of the course Mathematical Physics (Site not responding. Last check: 2007-11-05)
		Covariant derivative; covariant derivative along a curve: existence and uniqueness; component expression of the covariant derivative and of the covariant derivative along a curve.
		Curvature; definition of the Riemann tensor; relation of the Riemann tensor with the second covariant derivatives of a vector field; components of the Riemann tensor in a coordinate basis; symmetry properties of the Riemann tensor; Ricci tensor.
		Derivations of Lorentz transformation laws starting from the special relativity principle and from the law of propagation of light in vacuo.
	www-dft.ts.infn.it /~ansoldi/Didactics/Teaching/MatPhys/HTML/ProgEng.html (504 words)

	Covariant derivation (Site not responding. Last check: 2007-11-05)
		The same rules are valid for this as for the partial derivative, however the covariant derivatives with respect to upper indices are not converted to lower indices automatically and the covariant derivative of the metric tensor and that of the the Levi-Civita tensor are equal to zero.
		The non-operator notation for the partial and covariant derivatives, which is frequently used in the literature, is applied in the output as shown by the examples above.
		In this convention the partial derivative is denoted by a comma, the covariant derivative by a semicolon.
	www.kfki.hu /(hu)/cnc/szhkpub/riccir/node14.html (120 words)

	Re: Covariant Derivative (Site not responding. Last check: 2007-11-05)
		Covariant > derivative (defined simply by parallel transport) is not a > derivation.
		In that case it seems that the action can still be made gauge invariant without requiring the covariant derivative to be a derivation.
		But in non-abelian theories the gauge field product has to be commutation in order that the covariant derivative is a derivation.
	www.lns.cornell.edu /spr/2002-04/msg0040883.html (227 words)

	442 (Site not responding. Last check: 2007-11-05)
		Lecture 3: The covariant derivative of a contavariant vector, the connection.
		The covariant derivative of a general tensor using Leibnitz rule.
		The square root of g multiplied by the covariant derivative is the derivative multiplied by the sqyare root of g.
	www.maths.tcd.ie /~houghton/TEACHING/442/442lectures.html (803 words)

	ipedia.com: Covariant derivative Article (Site not responding. Last check: 2007-11-05)
		Occasionally the term "covariant derivative" refers to a derivative of sectionss of a general vector bundle along a tangent vector of the base; see subsection "Vector bundles" in "Connection form.
		Given a function, the covariant derivative coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by and by.
		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 and are any two tensors:
	www.ipedia.com /covariant_derivative.html (1203 words)

	Gravity: Nilpotent
		Equations involving the ordinary derivative depend on the coordinate system, whereas an equation about tensors that involves the covariant derivative, if true for one coordinate system, is true for all.
		If the value represents something at a point that doesn't move with the argument point being differentiated, then covariant corrections do not occur, and the conclusion is the same as before: the covariant and ordinary derivative are interchangeable.
		So the covariant corrections on the #2 subscript can be ignored; only the corrections on the #1 subscript survive.
	www.math.ucla.edu /~jimc/klein_h/d2=0.html (534 words)

	[No title]
		One purpose of this notation is to distinguish the covariant derivative operations that are defined directly from those that must be constructed.
		Instead, we are \emph{imposing} the rule and using it to extend the definition of the covariant derivative.
		The covariant derivative $D_{u}$ is acting on a form-field that has the tangent vector field $v$ as its argument.
	www.people.vcu.edu /~rgowdy/phys591/rap/diffgeom.rap (4081 words)

	Rhett Herman
		They must be assigned no symmetries: These new tensors will represent 1, 2, 3, 4, 5, and 6 gauge covariant derivatives of the scalar "a", a scalar which depends on the electromagnetic (gauge) field present in the spacetime.
		These first definitions are done by hand in that they simply commute the gauge covariant derivative on the final two indices of the tensors ad1, ad2, etc. These are printed out just to see the results.
		Finally, replace the covariant derivatives of the tensor ad2_{abc} and the scalar a{c} with the generic tensors ad3_{abc} and ad1{c}, respectively.
	www.runet.edu /~rherman/mathtensor/gauge.html (847 words)

	Covariant Differentiation (Site not responding. Last check: 2007-11-05)
		And he misinterpreted this identity to mean the vanishing of the covariant derivatives, cf.
		In Section 4 it is shown how the covariant derivatives of vector components have to be calculated.
		This is extended to covariant differentiation of covectors (1-forms) by means of the somewhat deeper concept of covectors (cf.
	www.mathematik.tu-darmstadt.de /~bruhn/covar_deriv.htm (2405 words)

	Covariant Derivative (Site not responding. Last check: 2007-11-05)
		Covariant derivative expansion of the Yang-Mills effective action at high temper...
		On the symmetry classes of the first covariant derivatives of tensor fields -- f...
		Covariant derivative expansion of fermionic effective action at high temperature...
	www.scienceoxygen.com /math/523.html (141 words)

	General Relativity (Site not responding. Last check: 2007-11-05)
		The first term on the right-hand side constitutes the ordinary derivative of the vector field, and the second term eliminates the derivative of the transformation coefficients.
		The differences between ordinary derivatives for different coordinate systems are taken into account by the metric connections, and the relevant information is contained in the covariant derivative.
		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)

	Gravity: Notation and Definitions
		The covariant derivative of f in direction i, defined below in terms of the Christoffel symbols.
		A ``covariant tensor'' is an object whose coordinate representation is transformed similarly to a gradient (going from Y to X), by matrix multiplying it (going from Y to X) by the Jacobian of the map function.
		However, the ordinary derivative of a tensor can be separated into two parts; one is a tensor called the ``covariant derivative'' and the other is a matrix-type product of the original object with a set of functions called the ``Christoffel symbols''.
	www.math.ucla.edu /~jimc/klein_h/notation.html (1120 words)

	[No title]
		The theoretical bound on the highest order of covariant derivative required in the Karlhede approach is reduced to seven in the worst three cases (non-vacuum types N and D and vacuum type N) and five in all other cases (vacuum and non-vacuum types I, II, III and vacuum type D).
		In [4] and [5] it was shown that Karlhede's bound on the number of derivatives required for type D vacuum spacetimes could be reduced from 5 to 3 (the bound for the non-vacuum case was reduced to 4 in [6]).
		This was used in [11] to reduce the bound on the number of derivatives required to classify type N vacuum spacetimes from 7 to 5.
	www.maths.soton.ac.uk /staff/Vickers/karlhede.html (1029 words)

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