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Differential Geometry

Geometrical Methods of Mathematical Physics

Geometrical Methods of Mathematical Physics Lowest new price: $40.00
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Author: Bernard F. Schutz
Brand: Brand: Cambridge University Press

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.


  • Used Book in Good Condition

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Elementary Differential Geometry, Revised 2nd Edition, Second Edition

Elementary Differential Geometry, Revised 2nd Edition, Second Edition Lowest new price: $85.00
Lowest used price: $46.75
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Author: Barrett O'Neill

Written primarily for students who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Revised 2nd Edition, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

  • Over 36,000 copies sold worldwide
  • Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study
  • Extensive update of appendices on Mathematica and Maple software packages
  • Thorough streamlining of second edition's numbering system
  • Fuller information on solutions to odd-numbered problems
  • Additional exercises and hints guide students in using the latest computer modeling tools

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Morse Theory (Annals of Mathematic Studies AM-51)

Morse Theory (Annals of Mathematic Studies AM-51) Lowest new price: $61.61
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Author: John Milnor
Brand: John Milnor

One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory.

Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.?

Milnor has published five books with Princeton University Press.


  • Morse Theory Annals of Mathematic Studies AM 51

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Riemannian Geometry (Graduate Texts in Mathematics)

Riemannian Geometry (Graduate Texts in Mathematics) Lowest new price: $47.37
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Author: Peter Petersen
Brand: Ingramcontent

Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of  Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups.

Important revisions to the third edition include:

  • a substantial addition of unique and enriching exercises scattered throughout the text;
  • inclusion of an increased number of coordinate calculations of connection and curvature;
  • addition of general formulas for curvature on Lie Groups and submersions;
  • integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;
  • incorporation of several recent results about manifolds with positive curvature;
  • presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.

From reviews of the first edition:

"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."

―Bernd Wegner, ZbMATH


  • Riemannian Geometry Graduate Texts in Mathematics

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Information Geometry (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

Information Geometry (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics) Lowest new price: $96.19
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Author: Nihat Ay

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory.  Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated.

This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality.  Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo.

The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.

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Space-Time Algebra

Space-Time Algebra Lowest new price: $71.43
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Author: David Hestenes
Brand: David Hestenes

This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future.

At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.

These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible.

There is another aspect to Geometric Algebra, which is less tangible, and goes beyond questions of mathematical power and range. This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics. Examples of this are peppered throughout ‘Space-Time Algebra’, despite its short length, and some of them are effectively still research topics for the future.

From the Foreward by Anthony Lasenby


  • Space Time Algebra 2015

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Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics)

Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) Lowest new price: $37.01
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Author: John M. Lee

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

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Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics)

Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) Lowest new price: $78.62
Lowest used price: $65.00
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Author: Mikio Nakahara
Brand: Mikio Nakahara

Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.

Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.


  • Geometry Topology and Physics

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Ergodic Theory and Negative Curvature: CIRM Jean-Morlet Chair, Fall 2013 (Lecture Notes in Mathematics)

Ergodic Theory and Negative Curvature: CIRM Jean-Morlet Chair, Fall 2013 (Lecture Notes in Mathematics) Lowest new price: $50.04
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Brand: Ingramcontent

Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. 

The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.


  • Ergodic Theory and Negative Curvature CIRM Jean Morlet Chair Fall 2013 Lecture Notes in Mathematics

Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics)

Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) Lowest new price: $63.21
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Author: Sigurdur Helgason
Brand: Brand: American Mathematical Society

The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material.

Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebras over $\mathbf{R}$.

The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.

For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39.

Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.


  • Used Book in Good Condition

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