Modern differential geometry of curves and surfaces

Author: Alfred Gray
Publisher: Lewis Pub
Category: Mathematics
Page: 664
View: 6566

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This is the first advanced text/reference to explain the mathematics of curves and surfaces and describe how to draw the pictures illustrating them using Mathematica. Learn not only the classical concepts, ideas, and methods of differential geometry, but also how to define, construct, and compute standard functions. Also learn how to create new curves and surfaces from old ones. Material includes 150+ exercises, 175 Mathematica programs, and 225 geometric figures to develop the topics presented. A tutorial explaining how to use Mathematica in differential geometry is included as well. This text/reference is excellent for mathematicians, scientists, and engineers who use differential geometric methods and investigate geometrical structures.

Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition

Author: Elsa Abbena,Simon Salamon,Alfred Gray
Publisher: CRC Press
ISBN: 1351992201
Category: Mathematics
Page: 1016
View: 4918

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Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.

Differential Geometry of Curves and Surfaces

Author: Kristopher Tapp
Publisher: Springer
ISBN: 3319397990
Category: Mathematics
Page: 366
View: 1892

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This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships. Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it.

The Language of Shape

The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology
Author: S. Hyde,Z. Blum,T. Landh,S. Lidin,B.W. Ninham,S. Andersson,K. Larsson
Publisher: Elsevier
ISBN: 9780080542546
Category: Science
Page: 383
View: 7261

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This book develops the thesis that structure and function in a variety of condensed systems - from the atomic assemblies in inorganic frameworks and organic molecules, through molecular self-assemblies to proteins - can be unified when curvature and surface geometry are taken together with molecular shape and forces. An astonishing variety of synthetic and biological assemblies can be accurately modelled and understood in terms of hyperbolic surfaces, whose richness and beauty are only now being revealed by applied mathematicians, physicists, chemists and crystallographers. These surfaces, often close to periodic minimal surfaces, weave and twist through space, carving out interconnected labyrinths whose range of topologies and symmetries challenge the imaginative powers. The book offers an overview of these structures and structural transformations, convincingly demonstrating their ubiquity in covalent frameworks from zeolites used for cracking oil and pollution control to enzymes and structural proteins, thermotropic and lyotropic bicontinuous mesophases formed by surfactants, detergents and lipids, synthetic block copolymer and protein networks, as well as biological cell assemblies, from muscles to membranes in prokaryotic and eukaryotic cells. The relation between structure and function is analysed in terms of the previously neglected hidden variables of curvature and topology. Thus, the catalytic activity of zeolites and enzymes, the superior material properties of interpenetrating networks in microstructured polymer composites, the transport requirements in cells, the transmission of nerve signals and the folding of DNA can be more easily understood in the light of this. The text is liberally sprinkled with figures and colour plates, making it accessible to both the beginning graduate student and researchers in condensed matter physics and chemistry, mineralogists, crystallographers and biologists.

Curves and Surfaces

Author: Sebastián Montiel,Antonio Ros,Donald G. Babbitt
Publisher: American Mathematical Soc.
ISBN: 0821847635
Category: Mathematics
Page: 376
View: 8240

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This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces. Following the modern point of view on differential geometry, the book emphasizes the global aspects of the subject. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry. In order to study the global properties of curves and surfaces, it is necessary to have more sophisticated tools than are usually found in textbooks on the topic. In particular, students must have a firm grasp on certain topological theories. Indeed, this monograph treats the Gauss-Bonnet theorem and discusses the Euler characteristic. The authors also cover Alexandrov's theorem on embedded compact surfaces in $\mathbb{R}^3$ with constant mean curvature. The last chapter addresses the global geometry of curves, including periodic space curves and the four-vertices theorem for plane curves that are not necessarily convex. Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. It is suitable as the text for a first-year graduate course or an advanced undergraduate course.

IT Convergence and Services

ITCS & IRoA 2011
Author: James J. Park,Hamid Arabnia,Hang-Bae Chang,Taeshik Shon
Publisher: Springer Science & Business Media
ISBN: 9400725981
Category: Technology & Engineering
Page: 700
View: 8516

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IT Convergence and Services is proceedings of the 3rd FTRA International Conference on Information Technology Convergence and Services (ITCS-11) and the FTRA International Conference on Intelligent Robotics, Automations, telecommunication facilities, and applications (IRoA-11). The topics of ITCS and IRoA cover the current hot topics satisfying the world-wide ever-changing needs. The ITCS-11 will be the most comprehensive conference focused on the various aspects of advances in information technology convergence, applications, and services. The ITCS-11 will provide an opportunity for academic and industry professionals to discuss the latest issues and progress in the area of ITCS. In addition, the conference will publish high quality papers which are closely related to the various theories, modeling, and practical applications in ITCS. The main scope of ITCS-11 is as follows. Computational Science and Applications Electrical and Electronics Engineering and Technology Manufacturing Technology and Services Management Information Systems and Services Electronic Commerce, Business and Management Vehicular Systems and Communications Bio-inspired Computing and Applications IT Medical Engineering Modeling and Services for Intelligent Building, Town, and City The IRoA is a major forum for scientists, engineers, and practitioners throughout the world to present the latest research, results, ideas, developments and applications in all areas of intelligent robotics and automations. The main scope of IRoA-11 is as follows. Intelligent Robotics & Perception systems Automations & Control Telecommunication Facilities Artificial Intelligence The IRoA is a major forum for scientists, engineers, and practitioners throughout the world to present the latest research, results, ideas, developments and applications in all areas of intelligent robotics and automations. The main scope of IRoA-11 is as follows. Intelligent Robotics & Perception systems Automations & Control Telecommunication Facilities Artificial Intelligence

Differentialgeometrie von Kurven und Flächen

Author: Manfredo P. do Carmo
Publisher: Springer-Verlag
ISBN: 3322850722
Category: Technology & Engineering
Page: 263
View: 4725

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Inhalt: Kurven - Reguläre Flächen - Die Geometrie der Gauß-Abbildung - Die innere Geometrie von Flächen - Anhang

Manifolds and Differential Geometry

Author: Jeffrey Marc Lee
Publisher: American Mathematical Soc.
ISBN: 0821848151
Category: Mathematics
Page: 671
View: 1254

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Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.


Kurven - Flächen - Mannigfaltigkeiten
Author: Wolfgang Kühnel
Publisher: Springer-Verlag
ISBN: 3658006153
Category: Mathematics
Page: 284
View: 6404

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Dieses Buch ist eine Einführung in die Differentialgeometrie und ein passender Begleiter zum Differentialgeometrie-Modul (ein- und zweisemestrig). Zunächst geht es um die klassischen Aspekte wie die Geometrie von Kurven und Flächen, bevor dann höherdimensionale Flächen sowie abstrakte Mannigfaltigkeiten betrachtet werden. Die Nahtstelle ist dabei das zentrale Kapitel "Die innere Geometrie von Flächen". Dieses führt den Leser bis hin zu dem berühmten Satz von Gauß-Bonnet, der ein entscheidendes Bindeglied zwischen lokaler und globaler Geometrie darstellt. Die zweite Hälfte des Buches ist der Riemannschen Geometrie gewidmet. Den Abschluss bildet ein Kapitel über "Einstein-Räume", die eine große Bedeutung sowohl in der "Reinen Mathematik" als auch in der Allgemeinen Relativitätstheorie von A. Einstein haben. Es wird großer Wert auf Anschaulichkeit gelegt, was durch zahlreiche Abbildungen unterstützt wird. Bei der Neuauflage wurden einige zusätzliche Lösungen zu den Übungsaufgaben ergänzt.

Elementary Differential Geometry

Author: A.N. Pressley
Publisher: Springer Science & Business Media
ISBN: 1848828918
Category: Mathematics
Page: 474
View: 3179

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Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via ul

Differential Geometry of Manifolds

Author: Stephen T. Lovett
Publisher: CRC Press
ISBN: 1439865469
Category: Mathematics
Page: 440
View: 985

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From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra—topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra. Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

A First Course in Differential Geometry

Surfaces in Euclidean Space
Author: Lyndon Woodward,John Bolton
Publisher: Cambridge University Press
ISBN: 1108600913
Category: Mathematics
Page: N.A
View: 3345

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Differential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the Gauss–Bonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercises to test students' understanding of the material, and ends with a supplementary chapter on minimal surfaces that could be used as an extension towards advanced courses or as a source of student projects.

Lectures on Differential Geometry

Author: Iskander Asanovich Taĭmanov
Publisher: European Mathematical Society
ISBN: 9783037190500
Category: Mathematics
Page: 211
View: 7693

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This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. The book is based on lectures the author held repeatedly at Novosibirsk State University. It is addressed to students as well as to anyone who wants to learn the basics of differential geometry.

Differential Geometry and Its Applications

Author: John Oprea
Publisher: MAA
ISBN: 9780883857489
Category: Mathematics
Page: 469
View: 9421

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Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

Vibration and Damping in Distributed Systems

Author: Goong Chen,Jianxin Zhou
Publisher: CRC Press
ISBN: 9780849371615
Category: Mathematics
Page: 464
View: 7923

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Vibration and Damping in Distributed Systems, Volume I provides a comprehensive account of the mathematical study and self-contained analysis of vibration and damping in systems governed by partial differential equations. The book presents partial differential equations techniques for the mathematical study of this subject. A special objective of establishing the stability theory to treat many distributed vibration models containing damping is discussed. It presents the theory and methods of functional analysis, energy identities, and strongly continuous and holomorphic semigroups. Many mechanical designs are illustrated to provide concrete examples of damping devices. Numerical examples are also included to confirm the strong agreements between the theoretical estimates and numerical computations of damping rates of eigenmodes.

Geometric Differentiation

For the Intelligence of Curves and Surfaces
Author: I. R. Porteous
Publisher: Cambridge University Press
ISBN: 9780521002646
Category: Mathematics
Page: 333
View: 2592

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This is a revised version of the popular Geometric Differentiation, first edition.


Publisher: PHI Learning Pvt. Ltd.
ISBN: 8120346505
Category: Mathematics
Page: 256
View: 1277

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Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, while trying to answer them using calculus techniques. The geometry of differentiable manifolds with structures is one of the most important branches of modern differential geometry. This well-written book discusses the theory of differential and Riemannian manifolds to help students understand the basic structures and consequent developments. While introducing concepts such as bundles, exterior algebra and calculus, Lie group and its algebra and calculus, Riemannian geometry, submanifolds and hypersurfaces, almost complex manifolds, etc., enough care has been taken to provide necessary details which enable the reader to grasp them easily. The material of this book has been successfully tried in classroom teaching. The book is designed for the postgraduate students of Mathematics. It will also be useful to the researchers working in the field of differential geometry and its applications to general theory of relativity and cosmology, and other applied areas. KEY FEATURES  Provides basic concepts in an easy-to-understand style.  Presents the subject in a natural way.  Follows a coordinate-free approach.  Includes a large number of solved examples and illuminating illustrations.  Gives notes and remarks at appropriate places.