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Topic: Calculus of variations

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Jesse Douglas

	Calculus of variations - Wikipedia, the free encyclopedia
		Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers.
		The theory of optimal control is a generalization of the calculus of variations.
		Fomin, S.V. and Gelfand, I.M. : Calculus of Variations, Dover Publ., 2000
	en.wikipedia.org /wiki/Calculus_of_variations (1229 words)

	PlanetMath: calculus of variations (Site not responding. Last check: 2007-11-06)
		The fundamental theorem of the calculus of variations is that for continuous functions
		This condition, one of the fundamental equations of the calculus of variations, is called the Euler-Lagrange condition.
		This is version 9 of calculus of variations, born on 2002-02-16, modified 2006-01-19.
	planetmath.org /encyclopedia/CalculusOfVariations.html (940 words)

	CALCULUS OF VARIATIONS - LoveToKnow Article on CALCULUS OF VARIATIONS (Site not responding. Last check: 2007-11-06)
		The calculus of variations arose from the attempts that were made by ~j,, mathematicians in the 17th century to solve problems olthe of which the following are typical examples.
		The transformation of the second variations of integrals of various types into forms in which their signs can be determined by inspection subsequently became one of the leading problems of the calculus of variations.
		The variation of the integral JF(x, y, y)dx is identical with the line integral of F taken round a contour consisting of the varied curve AQPB and the stationary curve AB, in the sense AQPBA.
	www.1911ency.org /V/VA/VARIATIONS_CALCULUS_OF.htm (5115 words)

	Mahalanobis
		Calculus of variations deals with functionals (functions of functions), as opposed to ordinary calculus which deals with functions of real numbers.
		Another motivating example for the calculus of variations: Around that time (in the 1960's) Fermat proposed that light travels between two points over the path that is the least time of all the paths (click here for an excellent explanation).
		Digging deeper: The basic first-order necessary condition in the calculus of variations is the Euler-Lagrange equation.
	mahalanobis.twoday.net /stories/202589 (376 words)

	AllRefer.com - calculus of variations (Mathematics) - Encyclopedia
		calculus of variations, branch of mathematics concerned with finding maximum or minimum conditions for a relationship between two or more variables that depends not only on the variables themselves, as in the ordinary calculus, but also on an additional arbitrary relation, or constraint, between them.
		For example, the problem of finding the closed plane curve of given length that will enclose the greatest area is a type of isoperimetric (equal-perimeter) problem that can be treated by the methods of the variational calculus; the solution to this special case is the circle.
		The calculus of variations was founded at the end of the 17th cent.
	reference.allrefer.com /encyclopedia/C/calcul-var.html (297 words)

	49: Calculus of variations and optimal control; optimization
		Calculus of variations and optimization seek functions or geometric objects which are optimize some objective function.
		Calculus of Variations -- generalities and applications to particle motion (to minimize action)
		A calculus of variations problem: find the curve of minimal length which joins two points and includes an area of 1.
	www.math.niu.edu /~rusin/known-math/index/49-XX.html (432 words)

	Calculus of variations: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-06)
		Calculus of variations is a field of mathematics mathematics quick summary:
		Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas....
		The key theorem of calculus of variations is the Euler-Lagrange equation[Click link for more facts about this topic].
	www.absoluteastronomy.com /encyclopedia/c/ca/calculus_of_variations.htm (1361 words)

	13.4.1.1 Calculus of variations
		Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths.
		The calculus of variations addresses a harder problem in which optimization occurs over a space of functions.
		In the calculus of variations, there are many different ``directions'' because of the uncountably infinite number of ways to construct a small variation function that perturbs the original function (the set of all variations is an infinite-dimensional function space; recall Example 8.5).
	msl.cs.uiuc.edu /planning/node698.html (594 words)

	Calculus of variations (Site not responding. Last check: 2007-11-06)
		In differential calculus one seems to vary a variable and in calc of variations one varies functions.
		I know I found variational calculus to have a very different feel from any of the topology stuff I took (it has been a long time though.).
		The real downfall of variational calculus though has probably been the computer, since just about any variational problem can be discretized and put as a simple minimization problem (for example expanding the function in Fourier series) for which there are all sorts of computa- tional approaches.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH129.HTM (245 words)

	Amazon.com: Calculus of Variations: Books: S. V. Fomin,I. M. Gelfand (Site not responding. Last check: 2007-11-06)
		An Introduction to the Calculus of Variations by Charles Fox
		Introduction To The Calculus Of Variations by Bernard Dacorogna
		The key idea behind classical mechanics is the calculus of variations, and classical mechanics variational formulations are the basis of quantum mechanics and quantum field theories be these based on the Schrodinger picture, the Dirac picture or the path integral picture.
	www.amazon.com /exec/obidos/tg/detail/-/0486414485?v=glance (1094 words)

	Mathematics 575 Calculus of Variations (Site not responding. Last check: 2007-11-06)
		A group of methods aimed to find `optimal' functions is called Calculus of Variations.
		The Calculus of Variations has been originated by Bernoulli, Newton, Euler; systematically developed beginning from XVIII century; it still attracts attention of mathematicians and it helps scientists and engineers to find the best possible solutions.
		Calculus of Variations with Applications to Physics and Engineering.
	www.math.utah.edu /~cherk/teach/calc-var.html (139 words)

	Open Directory - Science: Math: Calculus (Site not responding. Last check: 2007-11-06)
		Calculus and Physics Practice Exams - Practice exams for applied calculus and physics in PDF and HTML formats.
		Calculus Solutions - Reference site with a vast amount of information and example problems which are alphabetically listed by topic.
		The University of Minnesota Calculus Initiative - Offers calculus application examples for the mathematical properties of a rainbow, the fundamental theorem of calculus, methods of maximizing structural beams in a building, and modeling population growth.
	dmoz.org /Science/Math/Calculus (880 words)

	calculus of variations on Encyclopedia.com
		CALCULUS OF VARIATIONS [calculus of variations] branch of mathematics concerned with finding maximum or minimum conditions for a relationship between two or more variables that depends not only on the variables themselves, as in the ordinary calculus, but also on an additional arbitrary relation, or constraint, between them.
		, … , and derivatives of these, the object being to determine the dependent variables as functions of the independent variables such that the integral will be a maximum or minimum.
		Calculus and the teaching of intermediate microeconomics: results from a survey.
	www.encyclopedia.com /html/c1/calcul-var.asp (417 words)

	Software for the Course of Calculus of Variations (Site not responding. Last check: 2007-11-06)
		Abstract: In teaching the course of calculus of variations for students in mechanics at the Ural State University, a serious attention is paid to application of numerical methods in classical model problems.
		Ahiezer, N.I. Lectures on the calculus of variations.
		Anisimov, V. Course on the calculus of variations.
	home.ural.ru /~iagsoft/chchel.html (872 words)

	Citebase - On complexes related with calculus of variations (Site not responding. Last check: 2007-11-06)
		We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them.
		The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle.
		We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0105223 (629 words)

	Calculus of Variations
		This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory.
		Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition.
		The prerequisites are knowledge of the basic results from calculus of one and several variables.
	www.indiaplaza.com /books/pd.aspx?sku=0521642035 (192 words)

	MA591X Calculus of Variations (Site not responding. Last check: 2007-11-06)
		Calculus of variations deal with problems in physics, engineering and applied mathematics that are governed by maximum or minimum principles.
		It can be shown that the resource must be used to grow the body mass first then for the reproduction after certain age.
		Calculus of variations is closely related to many areas of applied mathematics, e.g., control theory, Hamilton-Jacobi's theory, and the finite element method.
	www4.ncsu.edu /~xblin/ma591x/outline.html (269 words)

	ESAIM: Control, Optimisation and Calculus of Variations (Site not responding. Last check: 2007-11-06)
		Calculus of variations: minimization problems, existence and regularity properties of minimizers and critical points, variational methods for differential equations, homogenization, multiscale problems and geometric measure theory.
		Optimisation theory and calculus of variations: linear, quadratic, nonlinear programming, large scale systems, stochastic or combinatorial optimization, numerical algorithms, variational methods...
		Published under the scientific responsibility of the Société de Mathématiques Appliquées et Industrielles (SMAI) and with the support of the Centre National de la Recherche Scientifique (CNRS).
	www.edpsciences.org /cocv (227 words)

	INTRODUCTION TO THE CALCULUS OF VARIATIONS
		The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving.
		Serves as an excellent introduction to the calculus of variations?
		“This book provides non-mathematics students with an easy way to grasp the basic idea of the calculus of variations, and its possible applications in their field of study.
	www.worldscibooks.com /mathematics/p361.html (436 words)

	Amazon.com: Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications: Books: Tom M. Apostol (Site not responding. Last check: 2007-11-06)
		Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus by Michael Spivak
		His multivariable chapters are well-illustrated, but calculus on R^n seems to be trivial once calculus on R is under your belt from a good introductory book like Larson/Hostetler/Edwards at a high-school pace.
		I agree with the other reviewers, if you're familiar with calculus and LA and want to learn more about each and their connections, this is the bible, however, if you're a newcomer to one or both, definitely learn each separately and more simply.
	www.amazon.com /exec/obidos/tg/detail/-/0471000078?v=glance (1622 words)

	Read This: Introduction to the Calculus of Variations
		Lebesgue and Tonelli soon joined in and, collectively, their approaches sired what are today called "the direct methods of the calculus of variations." Wasting no time, on p.
		This having been said, it should be noted that while Dacorogna advertises his book as "a concise and broad introduction" to the calculus of variations at an undergraduate and beginning graduate level, he does presuppose the reader to be able and willing to work hard and do battle with some serious analysis.
		But as a more sophisticated introduction to the calculus of variations it's a very beautiful treatment, and will reward the diligent reader with a solid introduction to a great and grand subject and to a lot of beautiful hard analysis.
	www.maa.org /reviews/Dacorogna.html (855 words)

	Calculus of Variations (Site not responding. Last check: 2007-11-06)
		The textbook is John L. Troutman, Variational Calculus and Optimal Control.
		There is also a great introduction to Calculus of Variations in Chapter 21 of Vol II of Feynmann's Lectures on Physics.
		Another real character that is very important in the Calculus of Variations is the Irish mathematician William Rowan Hamilton -- see also here.
	www.math.gatech.edu /~morley/6582.html (239 words)

	THE CALCULUS OF VARIATIONS AND FUNCTIONAL ANALYSIS
		This is a book for those who want to understand the main ideas in the theory of optimal problems.
		It provides a good introduction to classical topics (under the heading of "the calculus of variations") and more modern topics (under the heading of "optimal control").
		It employs the language and terminology of functional analysis to discuss and justify the setup of problems that are of great importance in applications.
	www.worldscibooks.com /mathematics/5374.html (313 words)

	Software for the Calculus of Variations (Site not responding. Last check: 2007-11-06)
		Ivanov A.G. Keywords: Calculus of variations, educational software, numerical methods.
		In teaching the course of calculus of variations for students in mechanics at the Ural State University, a serious attention is paid to application of numerical methods in classical model problems.
		Krasnov M.A., Makarenko G.I., Kiselev A.I. Calculus of Variations, Nauka, Moscow, 1973 (in Russian).
	home.ural.ru /~iagsoft/chel1.html (500 words)

	Basic calculus of variations., Edward Silverman
		Remarks on : Remarks on the paper: ``Basic calculus of variations''..
		[2] Lamberto Cesari, An existence theorem of calculus of variations on parametric surfaces, Amer.
		[10] L. Turner, The direct method in the calculus of variations, Purdue Thesis, 1957.
	projecteuclid.org /getRecord?id=euclid.pjm/1102723676 (154 words)

	The Columbia Encyclopedia, Sixth Edition: calculus of variations@ HighBeam Research (Site not responding. Last check: 2007-11-06)
		Read the article after viewing a brief ad from our sponsor – no registration required.
		CALCULUS OF VARIATIONS [calculus of variations ] branch of mathematics concerned with finding maximum or minimum conditions for a relationship between two or more variables that depends not only on the variables themselves, as in the ordinary calculus, but also on an additional arbitrary relation, or constraint, between them.
		For example, the problem of finding the closed plane curve of given length that will enclose the greatest area is a type of isoperimetric (equal-perimeter) problem that can be treated by the methods of the variational calculus; the solution to this special case is...
	highbeam.com /doc/1E1:calcul-var/calculus+of+variations.html?... (190 words)

	Citebase - Leibniz Cohomology and the Calculus of Variations (Site not responding. Last check: 2007-11-06)
		In this paper we continue the investigation of Loday's Leibniz cohomology as a new invariant for differentiable manifolds.
		In particular the Leibniz coboundary of a k-tensor (in the sense of differential geometry) is computed in a local coordinate chart and then interpreted in terms of the calculus of variations.
		For example, the Leibniz coboundary of the metric two-tensor reduces to the first variation of arc length.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9808036 (281 words)

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