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Topic: Exterior covariant derivative

	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 (347 words)

	Derivative - Open Encyclopedia (Site not responding. Last check: 2007-10-18)
		The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change.
		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.
	open-encyclopedia.com /Derivative (2077 words)

	Read about Derivative at WorldVillage Encyclopedia. Research Derivative and learn about Derivative here! (Site not responding. Last check: 2007-10-18)
		The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x.
		If the velocity of a car is given, as a function of time; then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
		In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
	encyclopedia.worldvillage.com /s/b/Derivative (2031 words)

	Derivative (Site not responding. Last check: 2007-10-18)
		The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes.
		For example, referring to the two-dimensional graph of f, the derivative can also be regarded as the slope of the tangent to the graph at the point x.
		Given this geometrical interpretation, it is not surprising that derivatives can be used to determine many geometrical properties of graphs of functions, such as concavity or convexity.
	hallencyclopedia.com /Derivative (2293 words)

	Calculus (Site not responding. Last check: 2007-10-18)
		A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus, though this was not known in the West at the time, and he had no contact with Western scholars.
		Perhaps the first hint of the derivative one encounters in school is the formula Speed=Distance/Time for an object moving at constant speed.
		The derivative lies at the heart of the physical sciences.
	hallencyclopedia.com /Calculus (1465 words)

	Calculus - Open Encyclopedia (Site not responding. Last check: 2007-10-18)
		Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes of the function's arguments.
		The derivative of a function is directly relevant to finding its maxima and minima — because those are points at which the graph is (expected to be) flat.
		The modern version of calculus is known as real analysis; this consists of a rigorous derivation of the results of calculus as well as generalisations such as measure theory and functional analysis.
	open-encyclopedia.com /Calc (908 words)

	Re: Is the exterior covariant derivative an anti-derivation? (Site not responding. Last check: 2007-10-18)
		the exterior algebra), a graded derivation satisfies the graded Leibniz rule D(ab) = (Da)b + (-1)^ck a(Db) where a is a k-form and the derivation is of degree c (e.g.
		Since the exterior covariant derivative is of odd degree +1, if it satisfies the graded Leibniz rule it would be an "antiderivation".
		It's weird, most books I've found that treat the exterior covariant derivative in detail seem to also introduce the idea of a derivation, but none state whether D is a derivation or not...
	www.lns.cornell.edu /spr/2005-02/msg0067083.html (221 words)

	iqexpand.com (Site not responding. Last check: 2007-10-18)
		The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have.
		The andquot;simpleandquot; derivative of a function f with respect to x is denoted...
		Derivative The Derivative Definition of The Derivative The derivative of the function f (x) at the point is given and denoted by Some Basic Derivatives In the table below, u, v, and w are functions of the...
	derivative.iqexpand.com (2468 words)

	Derivative : search word (Site not responding. Last check: 2007-10-18)
		Category:Calculus In mathematics, the derivative of a function is one of the two central concepts of calculus.
		(The other one is the antiderivative, the inverse of the derivative.) The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes.
		right Derivatives are defined by taking the limit of the slope of secant lines as they approach a tangent line.
	www.searchword.org /de/derivative.html (2364 words)

	Exterior derivative -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the exterior derivative operator of (additional info and facts about differential topology) differential topology, extends the concept of the (A quality that differentiates between similar things) differential of a function
		and the Lie derivative of a general (additional info and facts about differential form) differential form is closely related to the exterior derivative.
		The differences are primarily notational; various identities between the two are provided in the article on (additional info and facts about Lie derivative) Lie derivatives.
	www.absoluteastronomy.com /encyclopedia/e/ex/exterior_derivative.htm (614 words)

	HODGE DUAL STRUCTURE OF THE EVANS THEORY (Site not responding. Last check: 2007-10-18)
		F = d ^ A. Here A is the scalar valued potential one-form, d ^ is the exterior derivative, F is the scalar valued electromagnetic field two-form, *F is the Hodge dual of F, J is scalar valued current three-form and mu0 is the permeability in vacuo in S.I. Eqn.
		Only the covariant exterior derivative transforms as a proper tensor, which is why it is always needed in non-Minkowski spacetimes.
		The exterior derivative d^ transforms as a two-form under general coordinate transformations but not as a vector under linear Lorentz transformations.
	www.aias.us /Comments/comments08162004b.html (465 words)

	[No title]
		In this scheme, the (pre) symplectic potential is transcribed as the boundary term that results from the `integration by parts' when implementing the variation of the Lagrangian, while the (pre)symplectic 2-form structure is its functional exterior derivative.
		By definition, the Lie derivative of $\varpi$ by $\delta_{\phi}{}$ vanishes while finding a symplectic potential $\vartheta$ which is indeed Lie-dragged by $\delta_{\phi}$ through the identity \eqa \pounds_{\delta_{\phi}}\vartheta\equiv\delta_{\phi}\hook\varpi+ \delta[\vartheta(\delta_{\phi})]=0,\label{eq:canLie} \eeqa implies that (modulo an additive constant) $H=\vartheta(\delta_{\phi}{})$ is the unique Hamiltonian generating the canonical transformation.
		This is achieved in a covariant manner by adding the invariant four form, $I_{EP}$ comprised of the Euler, $E_4$ and Chern-Pontryagin, $P_4$ invariants \begin{align} I_{EP}&=2F_{AB}\we F^{AB}\leftrightarrow F_{ab}\we F^{ab}+ i F_{ab}\we^\star F^{ab}\\ \intertext{whose variation contributes a boundary term} \vartheta_{EP}&=\delta\Gamma_{AB}\we F^{AB}.
	www.ma.utexas.edu /mp_arc/papers/03-416 (3333 words)

	Calculus
		This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph), but for intervals of time that are very short.
		The derivative is defined as a limit of a difference quotient.
		Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
	www.math.ucdavis.edu /~temple/MAT21B/SUPPLEMENTARY-ARTICLES/1HistoryOfCalc.html (1866 words)

	Covariant Derivative (Site not responding. Last check: 2007-10-18)
		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)

	Re: Is the exterior covariant derivative an anti-derivation?
		Subject: Re: Is the exterior covariant derivative an anti-derivation?
		Usually an antiderivation is like an "inverse" of a derivation, like the indefinite integral is to a derivative.
		I think what you want to say here is that D is a graded derivation.
	www.lns.cornell.edu /spr/2005-02/msg0067030.html (232 words)

	Cartan's Calculus: the interior product. (Site not responding. Last check: 2007-10-18)
		The exterior operations are defined on a variety for which the constraints of metric have not been defined.
		The exterior derivative is also independent of a connection.
		It is not equivalent to the tensor covariant derivative except under special circumstances.
	www.uh.edu /~rkiehn/ed3/ed3fre5.htm (271 words)

	Quotient rule - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-18)
		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)

	Curvature of Riemannian manifolds
		The articles Cartan connection and covariant derivative explain two different ways to introduce and calculate the curvature tensor.
		Curvature of Pseudo-Riemannian manifold can be expressed on the same way with only slight modifications.
		the curvature tensor measures anticommutativity of the covariant derivative.
	encyclopedia.codeboy.net /wikipedia/c/cu/curvature_of_riemannian_manifolds.html (892 words)

	How the Universe got its gravity and dark energy (Site not responding. Last check: 2007-10-18)
		J.A. Wheeler U(1) Maxwell EM A is the Cartan connection 1-form in internal space not space-time d is the exterior derivative dual to the co-form boundary operator on a multiply-connected manifold integration of the form (Betti numbers homology and all that).
		BUT the fact that exterior differential forms are not tensors, (as they are functionally well behaved without the constraint of tensor diffeomorphisms) always makes a tensor treatment of Cartan's ideas awkward and limited, and IMO somewhat useless for understanding topological aspects of physics.
		Moreover, from a thermodynamic point of view the use of a Covariant derivative is equivalent to saying that the process involved is adiabatic.
	www.talkaboutscience.com /group/sci.astro/messages/492714.html (732 words)

	giache (Site not responding. Last check: 2007-10-18)
		Working with a Lagrangian density which is invariant under general covariant transformations and using standard tools of the calculus of variations, we study the corresponding currents.
		In this section, first we characterize the generally covariant Lagrangian densities related to the configuration space of the metric-affine gravitation theory, then, in particular, we consider those whose dependence on the linear connections on X is through their curvature tensor.
		Thus the covariant character of the current (56) is manifest.
	www.sif.it /cimento/tocb/112.05/06/06.html (1555 words)

	Nonlinear Hodge Theory (Site not responding. Last check: 2007-10-18)
		Nonlinear Hodge theory introduces a quasilinear extension of the Hodge-Kodaira equations and can be used, for instance, to model the stationary flow of a compressible fluid, codimension-1 nonparametric minimal surfaces, the rotation of a nonrigid body, torsion in a rod, or capillary action.
		In all cases the potential is represented by a 0-form and the field by a d-closed 1-form, where d is the (flat) exterior derivative [SS].
		In this extension the role played by d-closed 1-forms in nonlinear Hodge theory is played by D-closed 2-forms, where D is the exterior covariant derivative.
	www.yu.edu /faculty/otway/Nonlinear.htm (444 words)

	[Maxima] exterior product & exterior derivative for the covariant antisymmetric tensors (Site not responding. Last check: 2007-10-18)
		In the second one I verify the consequence of the Cartan formula for the high order forms, the Lie derivative has to commute with the exterior one.
		The definitions of the exterior product and the exterior derivative are essentially relied on the properties of kdelta.
		The Lie derivative is defined though the Cartan formula.
	www.ma.utexas.edu /pipermail/maxima/2003/004240.html (281 words)

	DISCUSSION WITH BOB FLOWER (Site not responding. Last check: 2007-10-18)
		He says: "you can’t take any sort of covariant derivative of the spin connection, since it’s not a tensor." However, in section 9.3 (after Eq.
		b) you believe that the covariant exterior derivative operator D/\ can be applied meaningfully to the spin connection, while Carroll says it can not.
		To clarify this point, it would be helpful for you to show the explicit component-wise equations whereby "D/\ is defined as an operator which is identical to, and has the same effect as, the covariant exterior derivative" when applied to the spin connection.
	www.aias.us /Comments/comments01272005.html (437 words)

	NTU Info Centre: Calculus (Site not responding. Last check: 2007-10-18)
		Your speed (a derivative) in a car tells you about your change in location, relative to changes in time.
		In mathematical language, this is an example of "taking a limit." More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable.
		It is directly relevant to finding the maxima and minima of a function — because at those points the graph is flat.
	www.nowtryus.com /article:Calculus (1720 words)

	[No title]
		In both gravitation and electrodynamics the use of a covariant derivative implies that the Evans theory is intrinsically gauge invariant.
		The reason is that local gauge invariance in relativity theory is defined as the replacement of the ordinary derivative by a covariant derivative {12, 13).
		In the language of differential geometry this is the covariant exterior derivative D^ {9}, without which the geometry of spacetime is incorrectly defined.
	www.mathematik.tu-darmstadt.de /~bruhn/PHYSICAL_OPTICS1.doc (2853 words)

	Implicit function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-18)
		This creates two derivatives, one for y > 0 and another for y < 0.
		the condition on F can be checked by means of partial derivatives.
		For the important generalisation to functions of several variables, see implicit function theorem.
	xahlee.org /_p/wiki/Implicit_function.html (369 words)

	Curvature form
		here stands for exterior derivative, is the Lie bracket and D denotes the exterior covariant derivative\nMore precisely,
		If is a fiber bundle with structure group G one can repeat the same for\nthe associated principal G bundle.
		here D denotes the exterior covariant derivative and the torsion.
	encyclopedia.codeboy.net /wikipedia/c/cu/curvature_form.html (283 words)

	Citebase - Dirac Sigma Models
		Covariant curvature and symmetries of Lie algebroid gauge theories.
		This derivation is a variant of some ideas proposed recently by Douglas.
		It is shown how the deformation of the superconformal generators on the string's worldsheet by a nonabelian super-Wilson line gives rise to a covariant exterior derivative on loop space coming from a nonabelian 2-form on target space.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0411112 (938 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		- exterior form or covariant tensor &eta is an involutive automorphism of the exterior algebra or the covariant tensor algebra.
		- exterior form or covariant tensor &xi is an involutive anti-automorphism of the exterior algebra or the covariant tensor algebra.
		If [] is put around the name, then explicit evaluation of exterior derivatives of the ambient cobase forms and all commutators of ambient base vectors will be supressed.
	www.lancs.ac.uk /depts/spc/staff/chtw/Manifolds8.txt (8273 words)

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