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Topic: Exterior derivative

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	Cartan's Calculus: The exterior derivative
		This concept is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional exterior algebra subspace.
		What is remarkable is that the operational definition of the exterior derivative works on any p-form carrying it into a p+1 form.
		In the Cartan exterior calculus, the operations are the same and are represented by the same exterior derivative, but the operation operates on different algebraic subspaces.
	www.uh.edu /~rkiehn/ed3/ed3fre4.htm (308 words)

	Exterior derivative
		Exterior derivative extends the concept of the differential[?] of a function to differential forms of higher degree.
		Exterior derivative of a differential form of degree k is a differential form of degree k+1.
		For example, in 3 dimensional Euclidean space, exterior derivative of a 1-form corresponds to curl and exterior derivative of a 2-form corresponds to divergence.
	www.ebroadcast.com.au /lookup/encyclopedia/ex/Exterior_derivative.html (173 words)

	Derivative - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-18)
		In mathematics, the derivative of a function is one of the two central concepts of calculus.
		The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes.
		Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
	encyclopedia.worldsearch.com /derivative.htm (2129 words)

	Exterior derivative - Wikipedia, the free encyclopedia
		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.
		The differences are primarily notational; various identities between the two are provided in the article on Lie derivatives.
	www.wikipedia.org /wiki/Exterior_derivative (347 words)

	partial derivative - Article and Reference from OnPedia.com
		In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant.
		Equations involving an unknown function's partial derivatives are called partial differential equations and are ubiquitous throughout science.
		However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous.
	www.onpedia.com /encyclopedia/partial-derivative (517 words)

	Hodge dual - Wikipedia, the free encyclopedia
		In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors.
		The former space is n choose k-dimensional while the latter is n choose (n-k) dimensional, and by the symmetry of the binomial coefficients, these two dimensions are in fact equal.
		The combination of the * operator and the exterior derivative d generates the classical operators div, grad and curl, in three dimensions.
	en.wikipedia.org /wiki/Hodge_dual (1150 words)

	exterior derivative information. (Site not responding. Last check: 2007-10-18)
		Exterior derivative - definition of Exterior derivative in.
		In mathematics, the exterior derivative operator of differential topology, extends the concept..
		In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms..
	www.tag-generator.com /e/exterior_derivative.html (89 words)

	Extended exterior derivative and torsion
		Now, I let D be an extension of the exterior derivative and D_i be the standard covariant derivative in the ith direction, where D is defined as: Da = (-1)^p (D_i a) /\ dx^i, where the choice of factor (-1)^p should be obvious from the massaged version of da.
		I think that this extended exterior derivative D (which I probably re- invented) is a more natural operation than the standard exterior derivative d and its standard extension to vector-valued forms.
		D^2 is a derivation that satisfies the Leibniz rule D^2(a/\b) = (D^2a)/\b + a/\(D^2b) and i_T is an algebraic derivation i_T = 0 on 0-forms.
	www.lns.cornell.edu /spr/2001-01/msg0030535.html (703 words)

	Best Exterior Paint (Site not responding. Last check: 2007-10-18)
		Exterior algebra 1: ation, known as the '''wedge product''' or the '''exterior product''', is written as andand;.
		Exterior Angle Sum Conjecture 1: A Geometric Conjecture that states that the exterior angles of any polygon add to a sum of 360.
		Alchemy 69: the human body (the microcosm) is affected by the exterior world (the macrocosm), which includes the heavens 95: two of these qualities were interior and two were exterior.
	www.daikaiju.com /edge/24526-best%20exterior%20paint.html (506 words)

	Encyclopedia: Exterior-derivative (Site not responding. Last check: 2007-10-18)
		In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace.
		In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db).
		In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as...
	www.nationmaster.com /encyclopedia/Exterior_derivative (1518 words)

	Lie Derivative Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-18)
		In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M.
		The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M.
		The Lie derivative is closely related to the exterior derivative and thus to Elie Cartan's theory of differential forms.
	www.karr.net /encyclopedia/Lie_derivative (1154 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 (Click link for more info and facts about differential topology) differential topology, extends the concept of the (A quality that differentiates between similar things) differential of a function
		The (The choicest or most essential or most vital part of some idea or experience) kernel of d consists of the closed forms, and the (An iconic mental representation) image of the exact forms (cf.
		and the Lie derivative of a general (Click link for more info and facts about differential form) differential form is closely related to the exterior derivative.
	www.absoluteastronomy.com /encyclopedia/e/ex/exterior_derivative.htm (680 words)

	Search MathWorld (Site not responding. Last check: 2007-10-18)
		Derivative The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have.
		Partial Derivative Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.
		Logarithmic Derivative The logarithmic derivative of a function $f$ is defined as the derivative of the logarithm of a function.
	ciencias.unizar.es /~mdg/2003/10eduunivesq/laboratorio99/tercera%20parte/eric/search.html (518 words)

	Re: external derivative and connection
		Via an atlas you can define partial derivatives of the tensor field components with respect to the (local) coordinates, but this does not make much sense from the point of view of differential geometry, if the taken derivatives do not define objects, that are independent of the chart used to define them.
		The next important notion is, that with these totally anti-symmetric tensor fields and its exterior derivatives one can define integrals, which have also a coordinate independent meaning.
		If you antisymmetrise the covariant derivative, defined on tensor fields in a general manifold with an affine connection, then it might happen that this connection is not torsion free, but of course since the derivatives are covariant by construction also the antisymmetrised derivatives are.
	www.usenet.com /newsgroups/sci.physics.research/msg00588.html (579 words)

	Lie derivative -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		The Lie derivatives are represented by (Click link for more info and facts about vector field) vector fields, as (Click link for more info and facts about infinitesimal generator) infinitesimal generators of flows ((A person devoted to the active life) active (Click link for more info and facts about diffeomorphism) diffeomorphisms) on M.
		The Lie derivative can also be defined to act on general tensors, as developed in the next section.
		The Lie derivative is closely related to the (Click link for more info and facts about exterior derivative) exterior derivative and thus to (Click link for more info and facts about Elie Cartan) Elie Cartan's theory of (Click link for more info and facts about differential forms) differential forms.
	www.absoluteastronomy.com /encyclopedia/L/Li/Lie_derivative.htm (1451 words)

	Fundamental theorem of calculus - Open Encyclopedia (Site not responding. Last check: 2007-10-18)
		The derivative of this function is equal to the infinitesimal change in x per infinitesimal change in time (of course, the derivative itself is dependent on time).
		This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative.
		As the derivative of the antiderivative is the original function, $F'(c_i) = f(c_i) .$
	open-encyclopedia.com /Fundamental_theorem_of_calculus (932 words)

	PlanetMath: differential form
		Using local coordinates, the directional derivative operation can be expressed as
		The exterior derivative is a first-order differential operator
		exterior derivative, 1-form, exterior product, wedge product, interior product, tensorial
	planetmath.org /encyclopedia/Tensorial.html (326 words)

	Derivative Trading (Site not responding. Last check: 2007-10-18)
		The derivative of a function at a point measures the rate at which the function'svalue changes as the function's argument changes.
		That is, a derivative provides a mathematicalformulation of the notion of rate of change.
		Given this geometrical interpretation, itis not surprising that derivatives can be used to determine many geometrical properties of graphs of functions, such as concavity or convexity.
	www.elusiveeye.com /side19082-derivative-trading.html (459 words)

	Learn more about Maxwell's equations in the online encyclopedia. (Site not responding. Last check: 2007-10-18)
		The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities.
		Kaluza and Klein showed in the 1920's that the Maxwell's equations can be derived by extending general relativity into five dimensions.
		This strategy of using higher dimensions to unify different forces is an active area of research in particle physics.
	www.onlineencyclopedia.org /m/ma/maxwell_s_equations.html (1564 words)

	Land Rover UK - Exterior features
		All derivatives feature distinctive power vents, grille, and stylish headlights while a split tailgate design gives quick, convenient access to the rear.
		As with all Land Rover vehicles, The Range Rover is designed to drive through 45-degree gradients and traverse 35-degree side slopes, as well as wade through water 700mm deep.
		For the new TDV8 derivative, 19 inch wheels are standard, with a 20 inch wheel and tyre option also available.
	www.landrover.com /gb/en/Vehicles/New_Range_Rover/Exterior/Exterior_features.htm (459 words)

	Which Equation Explains the Market Cycle ? best calculus derivative rule (Site not responding. Last check: 2007-10-18)
		derivative of a quotient of two functions and apply this formula to several examples.
		Chain rule In calculus, the chain rule is a formula for the derivative of the...
		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.
	ascot.pl /th/Fourier1/calculus-derivative-rule.htm (423 words)

	Hirani, Anil N. (2003-05-09) Discrete exterior calculus. http://resolver.caltech.edu/CaltechETD:etd-05202003-095403
		The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated.
		Definitions are given for discrete versions of all the usual operators of exterior calculus.
		In many examples we find that the formulas derived from DEC are identitical to the existing formulas in the literature.
	etd.caltech.edu /etd/available/etd-05202003-095403 (379 words)

	Exterior Shading (Site not responding. Last check: 2007-10-18)
		1) " Exterior" -- In the context of Exterior Shading
		2) " Shading" -- In the context of Exterior Shading
		Shade is the blocking of sunlight (in particular direct sunshine) by any object, and also the shadow created by that object.
	www.lottery-news.net /dust14364-exterior_shading.html (572 words)

	Exterior derivative (Site not responding. Last check: 2007-10-18)
		It can be shown that exterior derivative is uniquely determined by these properties and its agreement with the differential on0-forms (functions).
		These correspondence reveals about a dozen formulas from vectorcalculus as merely special cases of the above three rules of exterior differentiation.
		The kernel of d consists of the closed forms, andthe image of the exact forms (cf.
	www.therfcc.org /exterior-derivative-43945.html (257 words)

	Finances Network
		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.
		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.
	financesnetwork.0catch.com (2306 words)

	Brick Exterior (Site not responding. Last check: 2007-10-18)
		Extreior Gateway Protocol 1: The '''Exterior Gateway Protocol''' ('''EGP''') is a routing pr 3: During the early days of the Internet, an Eterior Gateway Protocol, EGP version 3, was used to inte
		Exteroor Angle Sum Conjecture 1: A Geometric Conjecture that states that the Exterior angles of any polygon add to a sum of 360.
		Alchemy 69: the human body (the microcosm) is affected by the Exteror world (the macrocosm), which includes the heavens 95: two of these qualities were interior and two were Exteriur.
	www.thesonars.com /web/48212-brick.exterior.html (543 words)

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