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Topic: Tensor derivative

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Covariant derivative

	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.
		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 tensor in a fixed coordinate chart, but it is not a tensor in the sense that it is not invariant under coordinate changes.
	en.wikipedia.org /wiki/Covariant_derivative (1155 words)

	Tensor - Wikipedia, the free encyclopedia
		Tensors may be written down in terms of coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen frame of reference.
		In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain.
		Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
	en.wikipedia.org /wiki/Tensor (1888 words)

	PlanetMath: connection
		To compute a derivative of a function, one subtracts the value of the function at a point from the value at a nearby point.
		Since the notions of connection, parallel transport, and covariant derivative are so closely related, it is easy to translate propopsitions 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.
	planetmath.org /encyclopedia/Connection.html (2994 words)

	Lie derivative - Wikipedia, the free encyclopedia
		The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M.
		The Lie derivative can be defined for vector fields as well, however, by first defining the Lie bracket of a pair of vector fields X and Y, denoted [X,Y].
		The Nijenhuis-Lie derivative enjoys many algebraic properties similar to those of the Lie derivative, with one notable exception: it is not a derivation in the usual sense.
	en.wikipedia.org /wiki/Lie_derivative (1438 words)

	Covariant derivative - Encyclopedia, History, Geography and Biography
		In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transport, depends on the path along which the vector is translated.
		Therefore, the covariant derivative is not a tensor.
		Given a function $f$ , the covariant derivative $\nabla_{\mathbf v}f$ coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by ${\mathbf v}f$ and by $df({\mathbf v})$ .
	www.arikah.net /encyclopedia/Covariant_derivative (1397 words)

	Covariant derivative (Site not responding. Last check: 2007-10-21)
		In a curved space, such as the surface of the Earth, the translation is not well defined and its analog, parallel transport, is depending on the path of moving the vector.
		The covariant derivative of a covector field along a vector field v is agin a a covector field.
		The covariant derivative of a tensor field along a vector field v is agin a tensor field of the same type.
	www.sciencedaily.com /encyclopedia/covariant_derivative (1168 words)

	Re: external derivative and connection
		For a tensor derivative we need neighbourhood and some way of matching up tensors in the tangent and co-tangent spaces.
		That is the fundamental operation of the exterior derivative is in some general sense the derivative of a scalar field and hence no connection is needed to define what we mean by change.
		This is certainly apparent from the co-ordinate definition of exterior derivative.
	www.lns.cornell.edu /spr/2003-09/msg0054018.html (519 words)

	[No title]
		Tensor operations such as contraction or covariant differentiation are carried out by actually summing over repeated (dummy) indices with DO statements.
		Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.
		When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices.
	www.unf.edu /public/cap4630/kmartin/gradfall94/maxima/tensor/manual.txt (4543 words)

	Lie derivative - Encyclopedia, History, Geography and Biography
		In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M.
		where the tensor product symbol $\otimes$ is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.
		The Lie derivative is closely related to the exterior derivative and thus to Elie Cartan's theory of differential forms.
	www.arikah.net /encyclopedia/Lie_derivative (1127 words)

	sciforums.com - covariant derivative
		All tensors are tensor fields in that the tensor is defined throughout the space on which it is defined.
		A tensor is defined at a point P on a manifold according to certain criteria (e.g.
		I have a doubt: you have the stress-energy tensor of General Relativity, and then I have learned that the covariant derivative of the stress-energy tensor is zero.
	www.sciforums.com /showthread.php?t=34327 (537 words)

	Physics Help and Math Help - Physics Forums - Christoffel symbol as tensor
		That whole derivation that led to that is quite involved so I don't quite understand how Wald can say that the Christoffel symbols are tensors since normally one does not refer to them as tensors, in fact its commonly understood that they are not tensors.
		So maybe the confusion is that when he changes coordinates, in order to have a tensor, he would have to transform both derivative operators by the tensor law, whereas he is merely taking one of the coordinate derivative operators as his delta a, rather than taking whatever he gets when he transforms the primed one.
		For example, a Lorentz tensor is not a tensor in the normal sense of the term since the Lorentz tensor is only a tensor under a Lorentz transformation.
	www.physicsforums.com /printthread.php?t=40177 (2159 words)

	MathTensor Usage Messages
		Detg is the determinant of the metric tensor.
		LD[wp,w1] is the Lie derivative of the p-form wp with respect to the 1-form w1.
		MaxwellTexpression[la,lb] is the expression for the Maxwell stress tensor in terms of the Maxwell field tensor, MaxwellF.
	smc.vnet.net /usage.html (11090 words)

	THEORY MANUAL: 2. FORMULATION OF THE CONTINUUM PROBLEM, 2.1-2.2 (Site not responding. Last check: 2007-10-21)
		It includes fundamental equations derived from first principles (for laminar flow) as well as equations that represent approximations for common, real-world flow situations such as those involving turbulence and/or chemical reactions.
		Temporal derivative notation employs either a standard derivative operator or a dot (·) to represent a derivative with respect to time.
		In standard tensor notation, subscripted indices are used to identify the order and components of a tensor.
	www.shef.ac.uk /mecheng/staff/xyl/fidap/help/theory/th02_01-02.htm (2675 words)

	Consult von Karman Institute for Fluid Dynamics
		Figure 1 displays the location of the quadripole source terms (second spatial derivative of the Lighthill tensor) in the case of the fully expanded jet.
		It is observed that the major source of sound results from an oscillation of the first shock after the nozzle outlet whereas in the screech tone phenomenon, the source of sound is located at the interaction of the shear layer with the fourth or fifth shock.
		Although the physics of the screech tone was not well captured using the Navier stokes approach with random perturbations, the computed sound pressure levels are found to agree reasonably well with NASA data as shown in Fig.
	www.vki.ac.be /research/themes/aeros99/7.html (801 words)

	[No title]
		In both cases we assumed a metric-compatible connection and used projection tensors defined in terms of the perpendicular or normal to a surface or to a curve.
		The projected derivative is defined by taking the directional derivative of an object and then projecting the result back into the object's assigned subspace.
		Its projected derivative would then be just the ordinary derivative% \[ \hat{D}_{u}v=D_{u}v\text{.} \] \subsubsection{Projections of Covariant Derivatives} For a general, fully projected tensor, the covariant derivative can be decomposed into a set of quantities that belong to projected subspaces with a correction term for each index of the tensor.
	www.people.vcu.edu /~rgowdy/phys691/rap/pt.rap (4454 words)

	The covariant derivative (Site not responding. Last check: 2007-10-21)
		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)

	CSDC : Cartan's Calculus: the interior product. (Site not responding. Last check: 2007-10-21)
		The exterior derivative is also independent of a connection.
		It is not equivalent to the tensor Covariant derivative except under special circumstances.
		The primitive idea is that a Covariant derivative is a constrained or restricted process which when acting on a tensor produces another tensor.
	www22.pair.com /csdc/ed3/ed3fre5.htm (460 words)

	Quotient rule - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist.
		and h(x) ≠ 0; then, the rule states that the derivative of g(x) / h(x) is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator:
		For more information regarding the derivatives of trigonometric functions, see: derivative.
	xahlee.org /_p/wiki/Quotient_rule.html (263 words)

	Physics Help and Math Help - Physics Forums - Christoffel symbol as tensor
		In curved space, the derivative operator associated with the metric is identical with the coordinate derivative of any particular coordinate system at at most one point.
		They refer to objects which are tensors, under any transformation, but it's not useful to call them that because they refer to a different tensor in every (non-linear) frame.
		But its a tensor which is attached to a coordinate system and general tensors do not have this property.
	www.physicsforums.com /printthread.php?t=40177&pp=40 (3547 words)

	Tools of Tensor Calculus - Riemann manifolds
		In physics, the result of applying the Hodge star operator to a tensor is known as the dual of a tensor and their components are given by the above expression.
		The covariant derivative of a tensor T of kind (p,q) is another tensor of kind (p,q+1) designated by the symbol
		The covariant derivative receives different names in the bibliography as absolute derivative (Choquet pp.301) and gradient (Misner pp.208).
	baldufa.upc.es /xjaen/ttc/tutorial/metr.htm (985 words)

	The Society of Rheology: 71st Annual Meeting (Oct 1999) Paper GN3
		As a point of reference, the simple time derivative of a body field maps to space as Oldroyd's convected derivative, which retains all information of weight and kind arising from the originating body field even though Cartesian tensors have no sense of kind and are boolean in their ability to sense weight.
		Specifically, the intrinsic derivatives of the covariant and contravariant body metric tensors and their determinants are all zero.
		The intrinsic derivative of all body tensors possessing the same rank (independent of weight and kind) map into Caresian space as a single Jaumann rate that is only dependent on rank.
	www.rheology.org /sor99a/abstract.asp?PaperID=30 (330 words)

	Amazon.com: Vector and Tensor Analysis With Applications: Books: Richard A. Silverman (Site not responding. Last check: 2007-10-21)
		Its clear introduction to many delicate topics (covariant derivatives, metric tensors, geodesics, etc.) is still valuable even now when the differential form approach seems to have won the battle.
		All of the basic concepts of introductory Tensor Analysis were adequately dealt with in a relatively clear and concise way; however, the numerous errors, oversimplifications, and oversights was a constant source of annoyance and doubt.
		I have a solid foundation in vector analysis, but never felt comfortable with tensors and generalized coordinates, yet these are necessary for much of modern physics.
	www.amazon.com /exec/obidos/tg/detail/-/0486638332?v=glance (980 words)

	index.html (Mathematica 5.1 for Students - Personal Use Only) (Site not responding. Last check: 2007-10-21)
		The complete covariant 4-derivative of a contravariant 4-potential field strength tensor is the average amount of change symmetric tensor plus the deviation from the average amount of change antisymmetric tensor.
		The relativistic force F is written in terms of a derivative with respect to the interval τ.
		Since the derivative is with respect to spacetime, the effect of gravity could also be on the distribution of mass in space.
	world.std.com /~sweetser/quaternions/gravity/Lagrangian_to_tests/Lagrangian_to_tests.html (4779 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The field strength \ tensor term represents all the energy in a changing potential in flat or \ curved spacetime.
		The covariant derivative is a way to say the 4-derivative \ depends on how a metric changes.
		The \ relativistic force F is written in terms of a derivative with respect to the \ interval \[Tau].
	world.std.com /~sweetser/quaternions/notebooks/Lagrangian_to_tests.nb (3671 words)

	Tensorial: A Tensor Calculus Package -- from Mathematica Information Center
		The package should be useful both as an introduction to tensor calculations and for advanced calculations.
		There is complete freedom in the choice of symbols for tensor labels and indices.
		Partial, covariant, total, absolute and Lie derivative routines for any dimension and any order.
	library.wolfram.com /infocenter/MathSource/434 (185 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Kronecker tensor; Jacobian matrix of coordinate transform; 2D surfaces in Euclidean space; Vector fields on curved spaces; Metric properties on 2D surfaces; Euclidean verses Curved space; Manifolds; General covariant and contravariant vectors
		Generalization of the energy-momentum tensor to GR; Its divergence; Curvature tensor; Riemann tensor and its symmetry properties; Cyclic and Bianchi identities; Einstein's tensor and its divergence
		Gravitational redshift; Derivation of equations of motion for a particle moving in the Schwarzschild spacetime
	www.tcnj.edu /~wick/GRAstronomy161LectureSummary.html (813 words)

	Derivatives Portal
		Derivatives Portal is an online professional source of relevant documentation in the field of derivatives and risk.
		This portal was initiated by IMC Derivatives Foundation, a Dutch non-profit organisation and has the mission to promote the use and knowledge of derivatives among institutional investors and academic institutions.
		We provide short descriptions and links to the most frequently read articles, books, journals, papers, newspapers/magazines, websites and upcoming events around the field of derivatives trading and risk management.
	www.derivativesportal.org (268 words)

	The Derivative of Isotropic Tensor Functions, Elastic Moduli and Stress Rate: I. Eigenvalue Formulation -- Chen and Dui ...
		The Derivative of Isotropic Tensor Functions, Elastic Moduli and Stress Rate: I. Eigenvalue Formulation -- Chen and Dui 9 (5): 493 -- Mathematics and Mechanics of Solids
		The Derivative of Isotropic Tensor Functions, Elastic Moduli and Stress Rate: I. Eigenvalue Formulation
		We derive basis-free formulae for the derivative of general
	mms.sagepub.com /cgi/content/short/9/5/493 (143 words)

	Classical Fields (Site not responding. Last check: 2007-10-21)
		Classical fields are defined as tensor fields over a (spacetime) manifold, taking their values in certain vector bundles.
		The basic ingredients of such a description consist of the algebra of differential forms and their exterior derivatives.
		Classical Fields as Vector Bundle valued Tensor Fields over Spacetime
	www.math.ucdavis.edu /~manash/LecNotes/tensabs.html (119 words)

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