Differential Geometry and Its Applications

Author: John Oprea

Publisher: MAA

ISBN: 9780883857489

Category: Mathematics

Page: 469

View: 7186

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.
Release

Differential Geometry and Mathematical Physics

Author: John K. Beem,Krishan L. Duggal,American Mathematical Society,Canadian Mathematical Society

Publisher: American Mathematical Soc.

ISBN: 0821851721

Category: Mathematics

Page: 224

View: 4188

This book contains the proceedings of the Special Session, Geometric Methods in Mathematical Physics, held at the joint AMS-CMS meeting in Vancouver in August 1993. The papers collected here contain a number of new results in differential geometry and its applications to physics. The major themes include black holes, singularities, censorship, the Einstein field equations, geodesics, index theory, submanifolds, CR-structures, and space-time symmetries. In addition, there are papers on Yang-Mills fields, geometric techniques in control theory, and equilibria. Containing new results by established researchers in the field, this book provides a look at developments in this exciting area of research.
Release

Differential Geometry, Global Analysis, and Topology

Proceedings of a Special Session of the Canadian Mathematical Society Summer Meeting Held June 1-3, 1990

Author: Canadian Mathematical Society. Summer Meeting,Andrew J. Nicas

Publisher: American Mathematical Soc.

ISBN: 9780821860175

Category: Mathematics

Page: 185

View: 5602

This book contains the proceedings of a special session on differential geometry, global analysis, and topology, held during the Summer Meeting of the Canadian Mathematical Society in June 1990 at Dalhousie University in Halifax. The session featured many fascinating talks on topics of current interest. The articles collected here reflect the diverse interests of the participants but are united by the common theme of the interplay among geometry, global analysis, and topology. Some of the topics include applications to low dimensional manifolds, control theory, integrable systems, Lie algebras of operators, and algebraic geometry. Readers will appreciate the insight the book provides into some recent trends in these areas.
Release

Differential Forms with Applications to the Physical Sciences

Author: Harley Flanders

Publisher: Courier Corporation

ISBN: 0486139611

Category: Mathematics

Page: 240

View: 9204

A graduate-level text utilizing exterior differential forms in the analysis of a variety of mathematical problems in the physical and engineering sciences. Includes 45 illustrations. Index.
Release

Geometrical Methods of Mathematical Physics

Author: Bernard F. Schutz

Publisher: Cambridge University Press

ISBN: 9780521298872

Category: Mathematics

Page: 250

View: 2184

For physicists and applied mathematicians working in the fields of relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This book provides an introduction to the concepts and techniques of modern differential theory, particularly Lie groups, Lie forms and differential forms.
Release

Introduction to Topology and Geometry

Author: Saul Stahl,Catherine Stenson

Publisher: John Wiley & Sons

ISBN: 1118546148

Category: Mathematics

Page: 536

View: 7197

An easily accessible introduction to over threecenturies of innovations in geometry Praise for the First Edition “. . . a welcome alternative to compartmentalizedtreatments bound to the old thinking. This clearly written,well-illustrated book supplies sufficient background to beself-contained.” —CHOICE This fully revised new edition offers the most comprehensivecoverage of modern geometry currently available at an introductorylevel. The book strikes a welcome balance between academic rigorand accessibility, providing a complete and cohesive picture of thescience with an unparalleled range of topics. Illustrating modern mathematical topics, Introduction toTopology and Geometry, Second Edition discusses introductorytopology, algebraic topology, knot theory, the geometry ofsurfaces, Riemann geometries, fundamental groups, and differentialgeometry, which opens the doors to a wealth of applications. Withits logical, yet flexible, organization, the SecondEdition: • Explores historical notes interspersed throughout theexposition to provide readers with a feel for how the mathematicaldisciplines and theorems came into being • Provides exercises ranging from routine to challenging,allowing readers at varying levels of study to master the conceptsand methods • Bridges seemingly disparate topics by creating thoughtfuland logical connections • Contains coverage on the elements of polytope theory, whichacquaints readers with an exposition of modern theory Introduction to Topology and Geometry, Second Edition is anexcellent introductory text for topology and geometry courses atthe upper-undergraduate level. In addition, the book serves as anideal reference for professionals interested in gaining a deeperunderstanding of the topic.
Release

Aspects of Differential Geometry III

Author: Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

Publisher: Morgan & Claypool Publishers

ISBN: 1627058826

Category: Mathematics

Page: 159

View: 6910

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book III is aimed at the first-year graduate level but is certainly accessible to advanced undergraduates. It deals with invariance theory and discusses invariants both of Weyl and not of Weyl type; the Chern‒Gauss‒Bonnet formula is treated from this point of view. Homothety homogeneity, local homogeneity, stability theorems, and Walker geometry are discussed. Ricci solitons are presented in the contexts of Riemannian, Lorentzian, and affine geometry.
Release

Differential Geometry: The Interface between Pure and Applied Mathematics

The Interface Between Pure and Applied Mathematics : Proceedings of a Conference Held April 23-25, 1986 with Support from the National Science Foundation

Author: Mladen Luksic,Clyde Martin

Publisher: American Mathematical Soc.

ISBN: 082185075X

Category: Mathematics

Page: 273

View: 6357

Normally, mathematical research has been divided into 'pure' and 'applied', and only within the past decade has this distinction become blurred. However, differential geometry is one area of mathematics that has not made this distinction and has consistently played a vital role in both general areas. The papers in this volume represent the proceedings of a conference entitled 'Differential Geometry: The Interface Between Pure and Applied Mathematics', which was held in San Antonio, Texas, in April 1986. The purpose of the conference was to explore recent exciting applications and challenging classical problems in differential geometry. The papers represent a tremendous range of applications and techniques in such diverse areas as ordinary differential equations, Lie groups, algebra, numerical analysis and control theory.
Release

Differential Geometry of Curves and Surfaces

Author: Thomas F. Banchoff,Stephen T. Lovett

Publisher: CRC Press

ISBN: 1482247372

Category: Mathematics

Page: 414

View: 2664

Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students’ geometric intuition through interactive computer graphics applets supported by sound theory. The book explains the reasons for various definitions while the interactive applets offer motivation for certain definitions, allow students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits. New to the Second Edition Reworked presentation to make it more approachable More exercises, both introductory and advanced New section on the application of differential geometry to cartography Additional investigative project ideas Significantly reorganized material on the Gauss–Bonnet theorem Two new sections dedicated to hyperbolic and spherical geometry as applications of intrinsic geometry A new chapter on curves and surfaces in Rn Suitable for an undergraduate-level course or self-study, this self-contained textbook and online software applets provide students with a rigorous yet intuitive introduction to the field of differential geometry. The text gives a detailed introduction of definitions, theorems, and proofs and includes many types of exercises appropriate for daily or weekly assignments. The applets can be used for computer labs, in-class illustrations, exploratory exercises, or self-study aids.
Release

Aspects of Differential Geometry I

Author: Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

Publisher: Morgan & Claypool Publishers

ISBN: 1627056637

Category: Mathematics

Page: 154

View: 8672

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.
Release

Using the Mathematics Literature

Author: Kristine K. Fowler

Publisher: CRC Press

ISBN: 9780824750350

Category: Language Arts & Disciplines

Page: 475

View: 3939

This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.
Release

Differential Geometry

The Interface Between Pure and Applied Mathematics : Proceedings of a Conference Held April 23-25, 1986 with Support from the National Science Foundation

Author: Mladen Luksic,Clyde Martin,W. F. Shadwick

Publisher: American Mathematical Soc.

ISBN: 9780821854075

Category: Mathematics

Page: 273

View: 9696

Normally, mathematical research has been divided into ``pure'' and ``applied,'' and only within the past decade has this distinction become blurred. However, differential geometry is one area of mathematics that has not made this distinction and has consistently played a vital role in both general areas. The papers in this volume represent the proceedings of a conference entitled ``Differential Geometry: The Interface Between Pure and Applied Mathematics,'' which was held in San Antonio, Texas, in April 1986. The purpose of the conference was to explore recent exciting applications and challenging classical problems in differential geometry. The papers represent a tremendous range of applications and techniques in such diverse areas as ordinary differential equations, Lie groups, algebra, numerical analysis, and control theory.
Release

Partial Differential Equations and Their Applications

Author: Canadian Mathematical Society. Seminar

Publisher: American Mathematical Soc.

ISBN: 0821806874

Category: Mathematics

Page: 315

View: 4248

This volume presents lectures given at the 1995 Annual Seminar of the Canadian Mathematical Society on Partial Differential Equations and Their Applications held at the University of Toronto in June 1995. The conference consisted of a combination of minicourses, invited presentations, and contributed talks. In this volume readers will find contributions on a variety of topics related to PDE, such as spectral asymptotics, harmonic analysis, differential operators in hyperbolic manifolds, applications to geometry, mathematical physics, hydrodynamics, and the interaction between theory and numerical methods in PDE.
Release

The Heart of Calculus

Explorations and Applications

Author: Philip M. Anselone,John W. Lee

Publisher: The Mathematical Association of America

ISBN: 0883857871

Category: Mathematics

Page: 245

View: 679

This book contains enrichment material for courses in first and second year calculus, differential equations, modeling, and introductory real analysis. It targets talented students who seek a deeper understanding of calculus and its applications. The book can be used in honors courses, undergraduate seminars, independent study, capstone courses taking a fresh look at calculus, and summer enrichment programs. The book develops topics from novel and/or unifying perspectives. Hence, it is also a valuable resource for graduate teaching assistants developing their academic and pedagogical skills and for seasoned veterans who appreciate fresh perspectives. The explorations, problems, and projects in the book impart a deeper understanding of and facility with the mathematical reasoning that lies at the heart of calculus and conveys something of its beauty and depth. A high level of rigor is maintained. However, with few exceptions, proofs depend only on tools from calculus and earlier. Analytical arguments are carefully structured to avoid epsilons and deltas. Geometric and/or physical reasoning motivates challenging analytical discussions. Consequently, the presentation is friendly and accessible to students at various levels of mathematical maturity. Logical reasoning skills at the level of proof in Euclidean geometry suffice for a productive use of the book.
Release

Differential Geometry

Bundles, Connections, Metrics and Curvature

Author: Clifford Henry Taubes

Publisher: OUP Oxford

ISBN: 0191621226

Category: Mathematics

Page: 312

View: 5504

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.
Release

Tensor Calculus

A Concise Course

Author: Barry Spain

Publisher: Courier Corporation

ISBN: 0486428311

Category: Mathematics

Page: 125

View: 7507

A compact exposition of the theory of tensors, this text also illustrates the power of the tensor technique by its applications to differential geometry, elasticity, and relativity. Explores tensor algebra, the line element, covariant differentiation, geodesics and parallelism, and curvature tensor. Also covers Euclidean 3-dimensional differential geometry, Cartesian tensors and elasticity, and the theory of relativity. 1960 edition.
Release

Books in Series

Author: N.A

Publisher: N.A

ISBN: 9780835221092

Category: Monographic series

Page: 1756

View: 4250

Release

Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Author: Józef Banaś,Mohammad Mursaleen

Publisher: Springer

ISBN: 8132218868

Category: Mathematics

Page: 315

View: 7909

This book deals with the study of sequence spaces, matrix transformations, measures of noncompactness and their various applications. The notion of measure of noncompactness is one of the most useful ones available and has many applications. The book discusses some of the existence results for various types of differential and integral equations with the help of measures of noncompactness; in particular, the Hausdorff measure of noncompactness has been applied to obtain necessary and sufficient conditions for matrix operators between BK spaces to be compact operators. The book consists of eight self-contained chapters. Chapter 1 discusses the theory of FK spaces and Chapter 2 various duals of sequence spaces, which are used to characterize the matrix classes between these sequence spaces (FK and BK spaces) in Chapters 3 and 4. Chapter 5 studies the notion of a measure of noncompactness and its properties. The techniques associated with measures of noncompactness are applied to characterize the compact matrix operators in Chapters 6. In Chapters 7 and 8, some of the existence results are discussed for various types of differential and integral equations, which are obtained with the help of argumentations based on compactness conditions.
Release