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# Topic: Differential geometry

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 Differential geometry and topology - Open Encyclopedia   (Site not responding. Last check: 2007-10-22) Differential geometry is the study of geometry using calculus. Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. open-encyclopedia.com /Differential_geometry   (982 words)

 Learn more about Differential geometry and topology in the online encyclopedia.   (Site not responding. Last check: 2007-10-22) Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces and other objects were considered as lying in a space of higher dimension (for example a surface in an ambient space of three dimensions). A special case of differential geometry is Riemannian manifolds (see also Riemannian geometry): geometrical objects such as surfaces which locally look like Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and tangent space, differentiability, and vector and tensor fields. A symplectic manifold is a differentiable manifold equipped with a closed 2-form (that is, a closed non-degenerate bi-linear alternating form). www.onlineencyclopedia.org /d/di/differential_geometry_and_topology.html   (892 words)

 Differential geometry Differential geometry is the study of curves and surfaces, and of more abstract objects called manifolds that are a generalization of these. Differential geometry is a very active field of modern mathematics, with connections to many other branches of mathematics. A knowledge of differential geometry will be useful to students interested in further study in a variety of topics, such as Lie groups, topology, dynamical systems and partial differential equations. www2.maths.unsw.edu.au /amsiss04/diffgeom.html   (386 words)

 differential geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05 branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. Differential geometry was founded by Gaspard Monge and C. Gauss in the beginning of the 19th cent. The importance of differential geometry may be seen from the fact that Einstein’s general theory of relativity is formulated entirely in terms of the differential geometry, in tensor notation, of a four-dimensional manifold combining space and time. www.bartleby.com /65/di/differn-ge.html   (521 words)

 AllRefer.com - differential geometry (Mathematics) - Encyclopedia differential geometry, branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates), although in the 20th cent. the methods of differential geometry have been applied in other areas of geometry, e.g., in projective geometry. reference.allrefer.com /encyclopedia/D/differn-ge.html   (161 words)

 The History of Curvature   (Site not responding. Last check: 2007-10-22) Thus, the growth of analytic geometry in the seventeenth century was stunted, and the explicit invention of curvature was preempted. Differential geometry started with vague definitions and simple concepts and developed into the well-oiled machine that it is today. Differential geometry and curvature were natural applications for the Calculus because they provided words to its music--practical applications (map making, light ray travel, etc...) to the theory. www.brown.edu /Students/OHJC/hm4/k.htm   (3531 words)

 PlanetMath: bibliography for differential geometry As the authors say in the introduction ``An understanding of what it means for solutions of systems of differential equations to exist and be unique is more imporant than an ability to crank out general solutions.'') Starting with the basics of set theory, the authors develop the basics of topology and linear algebra. Tensors, forms, integration and differentiation are all covered, as are curvature and geodesics. This is version 9 of bibliography for differential geometry, born on 2004-03-26, modified 2005-02-26. planetmath.org /encyclopedia/BibliographyForDifferentialGeometry.html   (630 words)

 EDGE EDGE aims to encourage and facilitate research and training in major areas of differential geometry, which is a vibrant and central topic in pure mathematics today. Symplectic geometry is a part of geometry where `almost-complex' methods already play a large role, and this area forms an integral part of the proposed research. Twistor methods give a correspondence between holomorphic geometry and low-dimensional conformal geometry, and also allow the use of complex methods in the study of quaternionic geometry, (parts of) gauge theory and integrable systems, all of which are subjects included in this research proposal. edge.imada.sdu.dk   (525 words)

 Differential geometry and topology at opensource encyclopedia   (Site not responding. Last check: 2007-10-22) It is an analog of symplectic geometry which works for odd dimensional manifolds. Finsler geometry has Finsler manifold as the main object of study, it is a differential manifold with Finsler metric, i.e. Riemannian geometry has Riemannian manifold as the main object of study, its smooth manifolds with an additional structure which makes them look infinitesimally like Euclidean space and therefore allow to generalise the notion from Euclidean geometry such as gradient of a function, divergence, length of curves and so on. wiki.tatet.com /Differential_topology.html   (895 words)

 Wulf Rossmann   (Site not responding. Last check: 2007-10-22) This is a collection of lecture notes which I put together while teaching courses on manifolds, tensor analysis, and differential geometry. A comment about the nature of the subject (elementary differential geometry and tensor calculus) as presented in these notes. Many will be horrified by the flood of formulas and indices which here drown the main idea of differential geometry (in spite of the author's honest effort for conceptual clarity). www.mathstat.uottawa.ca /Profs/Rossmann/Differential%2520Geometry%2520book.htm   (474 words)

 53: Differential geometry Differential geometry is the language of modern physics as well as an area of mathematical delight. Geometry of spheres is in the sphere FAQ. A metric in the sense of differential geometry is only loosely related to the idea of a metric on a metric space. www.math.niu.edu /~rusin/known-math/index/53-XX.html   (461 words)

 Shiing-Shen Chern -- famed mathematician Shiing-Shen Chern, the famed UC Berkeley mathematician whose name is immortalized in the "Chern classes" of differential geometry, died Friday in Tianjian, China. He had the greatest impact on global differential geometry and complex algebraic geometry, which are fundamental to many areas of mathematics and theoretical physics. A Chern class is a numerical invariant attached to complex manifolds -- an object of study in differential geometry. www.sfgate.com /cgi-bin/article.cgi?f=/chronicle/archive/2004/12/09/BAG9BA8CAR1.DTL   (557 words)

 Kids.net.au - Encyclopedia Differential geometry -   (Site not responding. Last check: 2007-10-22) Differential geometry is basically the study of geometry using calculus. It has many applications in physics, especially in the theory of relativity. The central objects of study are Riemannian manifolds (see also Riemannian geometry): geometrical objects such as surfaces which locally look like Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and tangent space, differentiability, and vector and tensor fields. www.kids.net.au /encyclopedia-wiki/di/Differential_geometry   (126 words)

 www.caressa.it: Mathematics The term Differential Geometry was used for the first time by Luigi Bianchi in his classical books Lezioni di geometria differenziale (1894), which, along with the Leçons sur la théorie générale des surfaces (1887-96) by Gaston Darboux, has been the standard treatise in the subject for many years. Actually Differential Geometry was born as the natural application of the analytical techniques developed in XVII and XVIII cent. This is an introduction to differential and riemannian geometry in a clear and elementary style, though stating all notions with precision and at a satisfactory level of generality. www.caressa.it /matematica-en.html   (5457 words)

 Open Questions: Geometry and Topology This was a "top down" or "wholistic" view of geometry, in that it did not seek to analyze geometric objects in terms of their constituent parts (such as points or lines). Differential equations of mathematical physics, such as Maxwell's equations, are efficiently expressed in a coordinate-independent way using the language of manifold theory. The maps establishing equivalence between differentiable manifolds are called diffeomorphisms, and the category is known as the category of differentiable manifolds, or alternatively, smooth manifolds. www.openquestions.com /oq-ma003.htm   (14549 words)

 Natural operations in differential geometry A systematic treatment of naturality in differential geometry requires to describe all natural bundles, and this is also one of the undertakings of this book. Even though Ehresmann in his original papers from 1951 underlined the conceptual meaning of the notion of an \$r\$-jet for differential geometry, jets have been mostly used as a purely technical tool in certain problems in the theory of systems of partial differential equations, in singularity theory, in variational calculus and in higher order mechanics. Further, some functors of modern differential geometry are defined on the category of fibered manifolds and their local isomorphisms, the bundle of general connections being the simplest example. www.emis.de /monographs/KSM/index.html   (1563 words)

 CS 286c/ACM 256 Discrete Differential Geometry: Theory and Applications   (Site not responding. Last check: 2007-10-22) This seminar/project class is geared towards helping participants understand concepts and methods from differential geometry, in particular for 2 and 3-manifolds, in a discrete rather than discretized setup. Discrete differential geometry aims to preserve selected structure when going from a continuous abstraction to a finite representation for computational purposes. For example, for a piecewise linear approximation ("mesh") of a surface one may define Gaussian curvature in such a way that important theorems are preserved in the discrete setting. www.cs.caltech.edu /courses/cs286c   (110 words)

 Differential manifolds   (Site not responding. Last check: 2007-10-22) An Introduction to Differential Geometry in Mathematical Physics... Application of Lie groups and differentiable manifolds to general methods for si... Math 546 Topology and Geometry of Manifolds Spring 2002... www.scienceoxygen.com /math/658.html   (130 words)

 The Math Forum - Math Library - Diffrntl Geom. Differential Geometry preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives. A short article designed to provide an introduction to differential geometry, the language of modern physics and an area of mathematical delight. Animations of differential geometry topics: images constructed by Seaman using the programs in Modern Differential Geometry of Curves and Surfaces with Mathematica (Second Edition) by Alfred Gray. mathforum.org /library/topics/differential_g   (2302 words)

 Outline of the course Geometry of SpaceTime   (Site not responding. Last check: 2007-10-22) Differentiable structure and differentiable manifold; (differentiable) functions and maps on/between manifolds; (differentiable) curves on manifolds. Differentiable partition of unity; existence of differentiable partitions of unity (statement without proof). form/tensor fields over an open set and along a curve; coordinate expression of a form/tensor field; differentiable form/tensor fields, characterization of differentiable form/tensor fields; space of differentiable form/tensor fields on a manifold; the differential of a map as a 1-form field on the manifold. www-dft.ts.infn.it /~ansoldi/Didactics/Teaching/SpaceTimeCourse/Web/ProgEng.html   (1059 words)

 Geometry and Topology, Department of Mathematics, UIUC The document Graduate Study in Geometry and Topology outlines the general areas of geometry and topology studied here and describes the advanced undergraduate and graduate courses that are offered regularly. Differential geometry, gauge theory, holomorphic vector bundles, moduli spaces. Differential geometry, foliation theory, gauge theory, moduli spaces, low dimensional geometry and topology, topological quantum field theory. www.math.uiuc.edu /GraduateProgram/researchmath/geomtop.html   (335 words)

 First year curriculum Differentiation: mean value theorem, Taylor's theorem and Taylor's series, partial differentiation and total differentiability of functions of several variables. Differentiability of complex functions, analytic functions, Cauchy-Riemann equations. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. www.math.upenn.edu /grad/1stYearGrad.html   (895 words)

 Guardian Unlimited | The Guardian | Maths holy grail could bring disaster for internet He is very reclusive, won't talk to the press, has shown no indication of publishing this as a paper, which you would have to do if you wanted to get the prize from the Clay Institute, and has shown no interest in the prize whatsoever," Dr Devlin said. We have good reason to assume it has been and within the next 12 months, in the inner core of experts in differential geometry, which is the field we are speaking about, people will start to say, yes, OK, this looks right. Differential geometry is the subject that is really underneath understanding everything about space and spacetime." www.guardian.co.uk /uk_news/story/0,3604,1298728,00.html   (1267 words)

 Integration of differential forms   (Site not responding. Last check: 2007-10-22) Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach... Residues and differential operators on schemes, Amnon Yekutieli... The Limits of Stochastic Integrals of Differential Forms, Terry Lyons, Lucretiu... www.scienceoxygen.com /math/387.html   (188 words)

 Exterior differential   (Site not responding. Last check: 2007-10-22) Classical geometries defined by exterior differential systems on higher frame bu... Exterior Differential Calculus and Aggregation Theory: A Presentation and Some N... The exterior differential form representation of the quasi-one-dimensional flow... www.scienceoxygen.com /math/711.html   (239 words)

 Conferences   (Site not responding. Last check: 2007-10-22) Algebraic Geometry, Commutative Algebra and Topology, Romania, September 22-26, 2002. Special Structures in Differential Geometry, University of Durham (United Kingdom), July 30 to August 9, 2001. Geometry and Physics of Branes, Villa Olmo, Como (Italy), May 7-11, 2001. www.maths.ox.ac.uk /geometry/conferences.shtml   (992 words)

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