Mathematical / Computational / Theoretical physics Books

Book SynopsisThe Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student''s Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section Table of Contents1. The Fourier and Laplace transforms; 2. Laplace transform examples; 3. Properties of the Laplace transform; 4. Applications of the Laplace transform; 5. The Z-transform; References; Index.

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Book SynopsisIn addition to his ground-breaking research, Nobel Laureate Steven Weinberg is known for a series of highly praised texts on various aspects of physics, combining exceptional physical insight with his gift for clear exposition. Describing the foundations of modern physics in their historical context and with some new derivations, Weinberg introduces topics ranging from early applications of atomic theory through thermodynamics, statistical mechanics, transport theory, special relativity, quantum mechanics, nuclear physics, and quantum field theory. This volume provides the basis for advanced undergraduate and graduate physics courses as well as being a handy introduction to aspects of modern physics for working scientists.Trade Review'By using the notion of fundamental constituents as the guiding historical and theoretical principle, Weinberg manages to lay the foundations of diverse disciplines (hydrodynamics, statistical mechanics, kinetic theory, thermodynamics, special relativity, quantum mechanics and even field theory) in less than 300 pages.' CERN Courier, Opinion Reviews'Whereas many textbooks forgo historical notes, Weinberg delights the reader by adding terse yet apt context to the physical concepts he introduces … It is as if he is imagining what students might be puzzled by and then solves those problems … everyone will want to have Foundations of Modern Physics on their bookshelf. There is always something new to be found in it, and - similar to having a conversation about physics with Weinberg - there is never a dull moment when reading it.' Melissa Franklin, Physics TodayTable of ContentsPreface; 1. Early atomic theory; 2. Thermodynamics and kinetic theory; 3. Early quantum theory; 4. Relativity; 5. Quantum mechanics; 6. Nuclear physics; 7. Quantum field theory: assorted problems; Bibliography; Author index; Subject index.

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Book SynopsisUpdating one of the most widely used introductory textbooks on Einstein's general relativity, this third edition includes the latest updates on gravitational waves, black holes, and cosmology. It introduces the science of relativity to final-year undergraduates and graduate students, requiring only a minimal background in mathematics.Trade ReviewPraise for the second edition: 'Bernard Schutz's textbook A First Course in General Relativity quickly became a classic, notable for its use of the geometrical approach to the subject, combined with a refreshing succinctness. Since its first publication in 1985, the field of general relativity has exploded … Schutz has done a masterful job of incorporating these new developments into a revised edition, which is sure to become a new 'classic'.' Clifford M. Will, McDonnell Center for the Space Sciences, Washington University, St. LouisPraise for the second edition: 'This new edition retains all of the original's clarity and insight into the mathematical foundations of general relativity, but thoroughly updates the accounts of the application of the theory in astrophysics and cosmology. The result is an indispensable volume and this new edition will no doubt become a classic text in its own right.' Mike Hobson, Cavendish Laboratory, University of CambridgePraise for the second edition: 'Schutz has updated his eminently readable and eminently teachable A First Course in General Relativity. This text will be appreciated by any upper-level undergraduate with an interest in cosmology, astrophysics, or experimentation in gravitational physics.' Richard Matzner, The Center for Relativity, University of Texas at AustinPraise for the second edition: ' … marvellous … very clear … I cannot recommend this book highly enough to any physicist who wants a good introduction to general relativity.' David Burton, The ObservatoryPraise for the first edition: 'Schutz has such mastery of the material that it soon becomes clear that one is in authoritative hands, and topics are selected and developed only to a point where they prove adequate for future needs.' The Times Higher Education SupplementPraise for the first edition: '… ought to inspire more physicists and astronomers to teach and learn the other half of the twentieth century's revolution in physics.' Foundations of PhysicsPraise for the first edition: 'The book is a goldmine of cleverly constructed problems and exercises (and solutions!).' NaturePraise for the first edition: '… provides the first step into general relativity for undergraduate students with a minimal background in mathematics.' Zentralblatt MATH'Several generations of students have benefitted from the first two editions of Professor Bernard Schutz' beautiful introductory textbook on tensor algebra, manifolds, physics in curved space times, and Einstein's field equations. Why another edition now? The answer is that, in the last years, precision measurements of stellar orbits around the central massive black hole in the Galactic Center, the detection of gravitational waves from in-spiraling binary black holes and neutron stars with LIGO, and the detection of the central 'radio wave shadow' of the supermassive black hole in the galaxy M87 have suddenly opened the magical world of strongly curved spacetime to precision experimental tests. These experiments and much more to come from ground- and space-based gravitational wave studies have started a renaissance of interest in Einstein's theory.' Reinhard Genzel, Max Planck Institute for Extraterrestrial Physics'Students and teachers of general relativity will welcome this new edition of Schutz' hugely popular text, significantly broadened to cover the astonishing discoveries of gravitational-wave astronomy and their implications. A pioneer of the geometrical approach to undergraduate-level teaching of GR, the book remains unmatched in its highly readable style. With vim and authority, Schutz leads his readers masterfully from mathematical foundations to the forefront of research in astronomy and cosmology, providing them with the tools to understand future discoveries. With this new edition, Schutz' classic text remains as fresh and relevant as ever.' Leor Barack, University of Southampton'An outstanding textbook on general relativity written with the author's customary clarity and in his engaging style. It includes not only the basics of general relativity, but also recent developments in the direct detection of gravitational waves. A clear exposition of the essential ideas and methods.' Rong-Gen Cai, Chinese Academy of Sciences'Professor Schutz' informal style bewitches the reader into absorbing profound and complex concepts effortlessly. Physics is explained in a lucid style with minimal mathematics, without compromising on rigour. The recent excitement in the field of gravitational waves and its implications for astronomy and cosmology is adeptly conveyed. This edition has been enriched with several more exercises which the student or the young researcher will find illuminating and instructive.' Sanjeev Dhurandhar, Inter University Centre for Astronomy and Astrophysics'When I first taught from this book in the 1980s, my students and I loved it for its unusual combination of clarity and brevity. This third edition is not quite as brief because so much has happened in the subject! But for an all-around text with clear writing and an engaging style, it is still top of the class.' Clifford Will, University of Florida'A First Course in General Relativity by Bernard Schutz is an outstanding introductory text on Einstein's theory of general relativity and offers an invaluable resource for students interested in understanding the formal and physical foundations of modern spacetime theory.' Karim Thebault, University of Bristol'As with its previous editions, this textbook provides a fantastically accessible introduction to the key physical concepts of general relativity and the formalism used by its practitioners. The third edition gives a much-needed update accounting for discoveries since the previous edition, with the chapters on gravitational waves in particular serving as outstanding tutorials for students who are interested in astronomical applications of this subject.' Scott Hughes, Massachusetts Institute of TechnologyTable of ContentsPreface to the third edition page; Preface to the second edition; Preface to the first edition; 1. Special relativity; 2. Vector analysis in special relativity; 3. Tensor analysis in special relativity; 4. Perfect fluids in special relativity; 5. Preface to curvature; 6. Curved manifolds; 7. Physics in a curved spacetime; 8. The Einstein field equations; 9. Fundamentals of Gravitational Radiation; 10. Spherical solutions for stars; 11. Schwarzschild geometry and black holes; 12. Gravitational wave astronomy; 13. Cosmology; Appendix A. Summary of linear algebra; References; Index.

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Book SynopsisThis modern text combines fundamental principles with advanced topics and recent techniques in a rigorous and self-contained treatment of quantum field theory.Beginning with a review of basic principles, starting with quantum mechanics and special relativity, students can refresh their knowledge of elementary aspects of quantum field theory and perturbative calculations in the Standard Model. Results and tools relevant to many applications are covered, including canonical quantization, path integrals, non-Abelian gauge theories, and the renormalization group. Advanced topics are explored, with detail given on effective field theories, quantum anomalies, stable extended field configurations, lattice field theory, and field theory at a finite temperature or in the strong field regime. Two chapters are dedicated to new methods for calculating scattering amplitudes (spinor-helicity, on-shell recursion, and generalized unitarity), equipping students with practical skills for research. AccesTrade Review'Quantum Field Theory: From Basics to Modern Topics, by François Gelis, is a very welcome addition to the canon of literature on quantum field theory, impressive both in its breadth and depth. It covers, in a succinct fashion, foundational material in the subject and then treats many more modern developments: effective field theories, anomaly matching, recursion relations for gauge and gravitational amplitudes, strong fields, and more.' Laurence Yaffe, University of Washington'Though there are many books on quantum field theory, I have found this book valuable for its readable treatment of a diverse selection of modern topics from a uniform viewpoint. Subjects introduced well in this book that are hard to find elsewhere include Schwinger-Keldysh and finite-temperature field theory, modern tools for scattering amplitudes, worldline methods, as well as effective field theory. The discussion is illustrated with a rich set of examples, mainly from high energy physics.' John McGreevy, University of California, San DiegoTable of ContentsPreface; 1. Basics of quantum field theory; 2. Peturbation theory; 3. Quantum electrodynamics; 4. Spontaneous symmetry breaking; 5. Functional quantization; 6. Path integrals for fermions and photons; 7. Non-Abelian gauge symmetry; 8. Quantization of Yang–Mills theory; 9. Renormalization of gauge theories; 10. Renormalization group; 11. Effective field theories; 12. Quantum anomalies; 13. Localized field configurations; 14. Modern tools for tree amplitudes; 15. Wordline formalism; 16. Lattice field theory; 17. Quantum field theory at finite temperature; 18. Strong fields and semi-classical methods; 19. From trees to loops; Further reading; Index.

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Book SynopsisTrade Review"Stimulating reading" --New ScientistTable of ContentsFundamental principles of theoretical physics; The Gibbs distribution; Ideal gases; Solids; Non-ideal gases; Solutions; Chemical reactions; Fluctuations; Surfaces.

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Book SynopsisWritten by two PhDs in nuclear engineering, this book includes practical examples drawn from a working knowledge of physics concepts. You'll learn how to use the Python programming language to perform everything from collecting and analyzing data to building software and publishing your results.

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Book Synopsis1. Mathematical Introduction.- 1.1. Linear Vector Spaces: Basics.- 1.2. Inner Product Spaces.- 1.3. Dual Spaces and the Dirac Notation.- 1.4. Subspaces.- 1.5. Linear Operators.- 1.6. Matrix Elements of Linear Operators.- 1.7. Active and Passive Transformations.- 1.8. The Eigenvalue Problem.- 1.9. Functions of Operators and Related Concepts.- 1.10. Generalization to Infinite Dimensions.- 2. Review of Classical Mechanics.- 2.1. The Principle of Least Action and Lagrangian Mechanics.- 2.2. The Electromagnetic Lagrangian.- 2.3. The Two-Body Problem.- 2.4. How Smart Is a Particle?.- 2.5. The Hamiltonian Formalism.- 2.6. The Electromagnetic Force in the Hamiltonian Scheme.- 2.7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations.- 2.8. Symmetries and Their Consequences.- 3. All Is Not Well with Classical Mechanics.- 3.1. Particles and Waves in Classical Physics.- 3.2. An Experiment with Waves and Particles (Classical).- 3.3. The Double-Slit Experiment with Light.- 3.4. Matter Waves (de Broglie Waves).- 3.5. Conclusions.- 4. The Postulatesa General Discussion.- 4.1. The Postulates.- 4.2. Discussion of Postulates I -III.- 4.3. The Schrödinger Equation (Dotting Your i's and Crossing your ?'s).- 5. Simple Problems in One Dimension.- 5.1. The Free Particle.- 5.2. The Particle in a Box.- 5.3. The Continuity Equation for Probability.- 5.4. The Single-Step Potential: a Problem in Scattering.- 5.5. The Double-Slit Experiment.- 5.6. Some Theorems.- 6. The Classical Limit.- 7. The Harmonic Oscillator.- 7.1. Why Study the Harmonic Oscillator?.- 7.2. Review of the Classical Oscillator.- 7.3. Quantization of the Oscillator (Coordinate Basis).- 7.4. The Oscillator in the Energy Basis.- 7.5. Passage from the Energy Basis to the X Basis.- 8. The Path Integral Formulation of Quantum Theory.- 8.1. The Path Integral Recipe.- 8.2. Analysis of the Recipe.- 8.3. An Approximation to U(t) for the Free Particle.- 8.4. Path Integral Evaluation of the Free-Particle Propagator.- 8.5. Equivalence to the Schrödinger Equation.- 8.6. Potentials of the Form V=a + bx + cx2 + d? + ex?.- 9. The Heisenberg Uncertainty Relations.- 9.1. Introduction.- 9.2. Derivation of the Uncertainty Relations.- 9.3. The Minimum Uncertainty Packet.- 9.4. Applications of the Uncertainty Principle.- 9.5. The Energy-Time Uncertainty Relation.- 10. Systems with N Degrees of Freedom.- 10.1. N Particles in One Dimension.- 10.2. More Particles in More Dimensions.- 10.3. Identical Particles.- 11. Symmetries and Their Consequences.- 11.1. Overview.- 11.2. Translational Invariance in Quantum Theory.- 11.3. Time Translational Invariance.- 11.4. Parity Invariance.- 11.5. Time-Reversal Symmetry.- 12. Rotational Invariance and Angular Momentum.- 12.1. Translations in Two Dimensions.- 12.2. Rotations in Two Dimensions.- 12.3. The Eigenvalue Problem of Lz.- 12.4. Angular Momentum in Three Dimensions.- 12.5. The Eigenvalue Problem of L2 and Lz.- 12.6. Solution of Rotationally Invariant Problems.- 13. TheHydrogen Atom.- 13.1. The Eigenvalue Problem.- 13.2. The Degeneracy of the Hydrogen Spectrum.- 13.3. Numerical Estimates and Comparison with Experiment.- 13.4. Multielectron Atoms and the Periodic Table.- 14. Spin.- 14.1. Introduction.- 14.2. What is the Nature of Spin?.- 14.3. Kinematics of Spin.- 14.4. Spin Dynamics.- 14.5. Return of Orbital Degrees of Freedom.- 15. Addition of Angular Momenta.- 15.1. A Simple Example.- 15.2. The General Problem.- 15.3. Irreducible Tensor Operators.- 15.4. Explanation of Some Accidental Degeneracies.- 16. Variational and WKB Methods.- 16.1. The Variational Method.- 16.2. The Wentzel-Kramers-Brillouin Method.- 17. Time-Independent Perturbation Theory.- 17.1. The Formalism.- 17.2. Some Examples.- 17.3. Degenerate Perturbation Theory.- 18. Time-Dependent Perturbation Theory.- 18.1. The Problem.- 18.2. First-Order Perturbation Theory.- 18.3. Higher Orders in Perturbation Theory.- 18.4. A General Discussion of Electromagnetic Interactions.- 18.5. Interaction of Atoms with Electromagnetic Radiation.- 19. Scattering Theory.- 19.1. Introduction.- 19.2. Recapitulation of One-Dimensional Scattering and Overview.- 19.3. The Born Approximation (Time-Dependent Description).- 19.4. Born Again (The Time-Independent Approximation).- 19.5. The Partial Wave Expansion.- 19.6. Two-Particle Scattering.- 20. The Dirac Equation.- 20.1. The Free-Particle Dirac Equation.- 20.2. Electromagnetic Interaction of the Dirac Particle.- 20.3. More on Relativistic Quantum Mechanics.- 21. Path IntegralsII.- 21.1. Derivation of the Path Integral.- 21.2. Imaginary Time Formalism.- 21.3. Spin and Fermion Path Integrals.- 21.4. Summary.- A.l. Matrix Inversion.- A.2. Gaussian Integrals.- A.3. Complex Numbers.Trade Review`An excellent text....The postulates of quantum mechanics and the mathematical underpinnings are discussed in a clear, succint manner.' - American Scientist, from a review of the First EditionTable of ContentsMathematical Introduction.- Review of Classical Mechanics.- All Is Not Well With Classical Mechanics. The Postulates-A General Discussion.- Simple Problems in One Dimension.- The Classical Limit.- The Harmonic Oscillator.- The Path Integral Formulation of Quantum Theory.- The Heisenberg Uncertainty Relations.- Systems with N Degrees of Freedom.- Symmetries and Their Consequences.- Rotational Invariance and Angular Momentum.- The Hydrogen Atom.- Spin.- Addition of Angular Momenta.- Variational and WKB Methods.- Time-Independent Perturbation Theory.- Time-Dependent Perturbation Theory.- Scattering Theory.- The Dirac Equation.- Path Integrals-II.- Appendix.- Answers to Selected Exercises.- Table of Constants.- Index

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Book SynopsisIn his monumental 1687 work, Philosophiae Naturalis Principia Mathematica, known familiarly as the Principia, Isaac Newton laid out in mathematical terms the principles of time, force, and motion that have guided the development of modern physical science. This is a modern translation based on the 1726 edition.

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Book SynopsisThis introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles. While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. The goal of this book is to present important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other. Trade Review“He has written a book about principal bundles in the classical sense which is of great interest in and of itself … . a textbook which can be used in an advanced one-year course or for self-learning. … the book is also interesting for a physicist, because one can find the geometric basis of many mathematical tools used in physics. … reviewer has greatly enjoyed reading the book and acknowledges the author’s bravery in writing another text on differential geometry!” (Fernando Etayo Gordejuela, Mathematical Reviews, November, 2015)“The present book deals with principle bundles and their relevance in physics with a ground work on differential geometry. … The book will be helpful to the graduate and under graduate students of mathematics and physics. It can also be an informative hand book of the researchers in differential geometry and physics.” (Uday Chand De, zbMATH 1321.53004, 2015)Table of ContentsIntroduction.- Basics of Manifolds.- Vector Bundles.- Vectors and Covectors.- Differential Forms.- Lie Derivatives.- Lie Groups.- Frobenius Theorem.- Principle Bundles.- Connections on Principle Bundles.- Curvature of a Connection.- Classical Electromagnetism.- Yang-Mills Theory.- Gauge Theory.- The Dirac Monopole.- Instantons.- What Next?.- Discussion of the Exercises.

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Book SynopsisThe book aims to present current knowledge concerning the propagation of electro­ magnetic waves in a homogeneous magnetoplasma for which temperature effects are unimportant. It places roughly equal emphasis on the radio and the . hydromagnetic parts of the electromagnetic spectrum. The dispersion properties of a magnetoplasma are treated as a function both of wave frequency (assumed real) and of ionization density. However, there is little discussion of propagation in a stratified medium, for of collisions is included only which reference may be made to Budden [1] . The effect in so far as this can be done with simplicity. The book describes how pulses are radiated from both small and large antennas embedded in a homogeneous magneto­ plasma. The power density radiated from a type of dipole antenna is studied as a function of direction of radiation in all bands of wave frequency. Input reactance is not treated, but the dependence of radiation resistance on wave frequency is described for the entire electromagnetic spectrum. Also described is the relation between beaming and guidance for Alfven waves.Table of Contents1. Elementary properties of a plasma.- Plasma.- Equations of drift motion.- Isothermal atmosphere in equilibrium.- Types of wave.- Effect of collisions.- The continuity equations.- 2. Maxwell’s equations.- Equations in terms of current and charge densities.- Equations in terms of electric moment per unit volume.- The exponential wave function.- The concept of a dispersion relation.- Calculation of the dispersion relation (electric current method).- Calculation of the dispersion relation (electric moment method).- 3. Isotropic plasma.- Mobility and conductivity of an isotropic plasma.- Susceptibility and dielectric constant of a collisionless isotropic plasma.- The plasma frequency.- Refractive index of a collisionless isotropic plasma.- Wave dispersion in a collisionless isotropic plasma.- Effect of collisions in an isotropic plasma.- Importance of ordered kinetic energy in a plasma.- Poynting’s theorem in a plasma.- The energy significance of the complex dielectric constant of an isotropic plasma.- 4. Alternating current in a magnetoplasma.- Mobility tensor for a magnetoplasma.- Conductivity tensor for a magnetoplasma.- Low-frequency conduction properties of an infinite homogenous magnetoplasma.- Low-frequency conduction properties of a slab of magnetoplasma.- Effect of plasma scale on wave propagation.- 5. General properties of phase propagation in a magnetoplasma.- Susceptibility tensor for a magnetoplasma.- Alternative expressions for the susceptibility tensor components in the absence of multiple ion species.- Dispersion relation for a magnetoplasma.- Elliptic polarization.- Alternative derivation of the dispersion relation for a magnetoplasma.- The radio and hydromagnetic approximations.- Effect of collisions in a magnetoplasma.- 6. General properties of group propagation in a magnetoplasma.- Frequency and angular spectra.- Velocity of a wave packet.- Relation between phase and group propagation.- Method for calculating group velocity in a magnetoplasma.- Formulae for group velocity in a magnetoplasma.- Beam radiation in a magnetoplasma.- 7. Propagation of phase along the imposed magnetic field.- Circular polarization.- The dispersion relation for longitudinal propagation.- Longitudinal Alfvén waves.- The violin-string approach to longitudinal Alfven waves.- The hydromagnetic approximation for longitudinal propagation.- The radio approximation for longitudinal propagation.- The Eckersley approximation for longitudinal propagation.- Comparison of approximations.- Pass and stop bands of frequency for longitudinal propagation.- Particle vibration for longitudinal propagation.- Plasma motion in a longitudinal Alfvén wave.- Longitudinal propagation in low-density and high-density magnetoplasmas.- Effect of collisions on longitudinal propagation.- Effect of an additional ion species on longitudinal propagation.- Pass and stop bands of ionization density for longitudinal propagation.- 8. Energy flow and group velocity for longitudinal propagation.- Electromagnetic energy density for longitudinal propagation.- Kinetic energy density for longitudinal propagation.- Energy flow and group velocity for longitudinal propagation.- Energy in a longitudinal Alfvén wave.- Faraday rotation for longitudinal Alfvén waves.- A resonator for longitudinal Alfvén waves.- The mode of operation of a hydromagnetic violin-string.- Freezing of the magnetic field in the plasma (longitudinal Alfven waves).- Energy in a longitudinal whistler wave in the band ?Mi ? ? ? ?Me.- A resonator for longitudinal whistler waves in the band ?Mi ? ? ? ?Me.- Freezing of the magnetic field in the electron gas (longitudinal whistler wave).- Solid-state plasmas.- 9. Propagation of phase transverse to the imposed magnetic field.- The O wave.- The X wave.- Superposition of the O and X waves.- Pass and stop bands of frequency for transverse propagation.- The hybrid resonant frequencies.- Transverse propagation in a low-density magnetoplasma.- Pass and stop bands of ionization density for transverse propagation.- Effect of collisions on transverse propagation.- 10. Elliptic polarization of the X wave for transverse propagation.- The electric ellipse for transverse propagation of the X wave.- Frequency dependence of the electric ellipse.- Particle vibration for transverse propagation of the X wave.- Plasma compressions and dilations for transverse propagation of the X wave.- Non-reciprocity.- 11. Energy behaviour of the X wave for transverse propagation.- Electromagnetic energy density for transverse propagation of the X wave.- Kinetic energy density for transverse propagation of the X wave.- Energy flow and group velocity for transverse propagation of the X wave.- A resonator for transverse Alfvén waves.- The mode of operation of a hydromagnetic organ-pipe.- Freezing of the magnetic field in the plasma (transverse Alfvén waves).- 12. Propagation at any angle to the imposed magnetic field..- The zeros in the frequency dispersion curves.- Nomenclature for the characteristic waves.- The cross-connection phenomenon for frequency dispersion curves.- Frequency dispersion curves for nearly transverse propagation.- Frequency dispersion curves for nearly longitudinal propagation.- The elliptic polarizations of the O and X waves at the plasma frequency.- Effect of an additional ion species on cross-connection phenomena.- The infinities in the frequency dispersion curves.- Permitted regions for the frequency dispersion curves.- The cross-connection phenomenon for ionization dispersion curves.- Permitted regions for the ionization dispersion curves.- Propagation into a magnetoplasma from free space.- 13. The radio approximation.- The radio approximation to the dispersion relation.- Frequency dispersion curves in the radio band.- Frequency dependence of elliptic polarization in the radio band.- Frequency dependence of the direction of group propagation in the radio band.- Variation in the angle of squint of a rotating broadside antenna in the radio band.- Dependence of refractive index on ionization density in the radio band.- Dependence of elliptic polarization on ionization density in the radio band.- Dependence of the direction of group propagation on ionization density in the radio band.- 14. The hydromagnetic approximation.- The hydromagnetic approximation to the dispersion relation.- Frequency dispersion curves in the hydromagnetic band.- Effect of ionic collisions in the hydromagnetic band.- The fit between the hydromagnetic and radio approximations.- Frequency dependence of elliptic polarization in the hydromagnetic band.- Frequency dependence of the tilts of the electronic and current ellipses in the hydromagnetic band.- Frequency dependence of the direction of group propagation in the hydromagnetic band.- Polar diagrams for group velocity in the hydromagnetic band.- Dependence of refractive index on ionization density in the hydromagnetic band.- Dependence of elliptic polarization on ionization density in the hydromagnetic band.- Dependence of the direction of group propagation on ionization density in the hydromagnetic band.- 15. The quasi-longitudinal and quasi-transverse approximations.- The transition angle between the quasi-longitudinal and quasi-transverse approximations.- The regions of validity for the first- order angular approximations.- Importance of avoiding angular approximations that upset an infinity of a refractive index.- The regions of validity for angular approximations of practical value.- Accuracy of ?2n/??p2 using angular approximations.- The quasi-transverse approximation when ? ? ?Mi.- The quasi-longitudinal approximation when ? ? Max(?N,?Me).- The quasi-longitudinal approximation when ?Mi ? ? < ?Me ? ?N.- Group propagation for the whistler wave when ??1 ? ? < ?Me ? ?N.- Comparison of the zero-order quasi-longitudinal approximation in the whistler band with the unapproximated formulae.- 16. Directional behaviour of group velocity in a magnetoplasma.- Group propagation of the X wave in the pass band ? > ?C2.- Group propagation of the O wave in the pass band ? > ?N.- Group propagation in the upper part of the whistler band [??1 < ? < Min(?N,?Me)].- Group propagation in the lower part of the whistler band (? ? ??1).- Group propagation of the O wave in the pass band ? < ?Mi.- Group propagation of the X wave in the pass band ?C1 ?Me.- Group propagation of the X wave in the pass band ?C1 < ? < ??2 when ?N < ?Me.- 17. The field of an antenna in a magnetoplasma.- Axes of coordinates.- Angular spectra of O and X waves.- The predominant directions of group and phase propagation in the far field.- The method of steepest descent.- Simplification of the notation.- The power density in the far field.- Use of the angle of phase propagation as an independent variable.- Radiation from a gaussian dipole in a homogeneous magnetoplasma.- A reference isotropic medium.- Radiation ?C2.- Radiation in the frequency band ?C2 > ? > ??2.- Radiation in the frequency band ?? > ? > ?N.- Radiation in the frequency band ?N > ? > ?C1.- Radiation in the frequency band ?Me > ? > ??1.- Radiation in the frequency band ??1 > ? > ?Mi.- Radiation in the frequency band ? < ?Mi.- Frequency dependence of radiation resistance.- The relation between beaming and guidance in a homogeneous magnetoplasma.- The relation between beaming and guidance for the whistler wave when ??1 < ? ? Min (?N, ?Me).- The relation between beaming and guidance for the O wave when ? ? ?Mi.- The relation between beaming and guidance for the combined O and X waves when ? ? ?Mi.- Effect of energy absorption on Alfvén guidance.- Symbols.- Index of subjects.

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Book SynopsisThis book – specifically developed as a novel textbook on elementary classical mechanics – shows how analytical and numerical methods can be seamlessly integrated to solve physics problems. This approach allows students to solve more advanced and applied problems at an earlier stage and equips them to deal with real-world examples well beyond the typical special cases treated in standard textbooks.Another advantage of this approach is that students are brought closer to the way physics is actually discovered and applied, as they are introduced right from the start to a more exploratory way of understanding phenomena and of developing their physical concepts.While not a requirement, it is advantageous for the reader to have some prior knowledge of scientific programming with a scripting-type language. This edition of the book uses Matlab, and a chapter devoted to the basics of scientific programming with Matlab is included. A parallel edition using Python instead of Matlab is also available.Last but not least, each chapter is accompanied by an extensive set of course-tested exercises and solutions.Table of Contents Introduction.- Getting started with programming.- Units and measurement.- Motion in one dimension.- Forces in one dimension.- Motion in two and three dimensions.- Forces in two and three dimensions.- Constrained motion.- Forces and constrained motion.- Work.- Energy.- Momentum, impulse, and collisions.- Multiparticle systems.- Rotational motion.- Rotation of rigid bodies.- Dynamics of rigid bodies.- Proofs.- Solutions.- Index.

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Book SynopsisSpecial relativity is the basis of many fields in modern physics: particle physics, quantum field theory, high-energy astrophysics, etc. This theory is presented here by adopting a four-dimensional point of view from the start. An outstanding feature of the book is that it doesn’t restrict itself to inertial frames but considers accelerated and rotating observers. It is thus possible to treat physical effects such as the Thomas precession or the Sagnac effect in a simple yet precise manner. In the final chapters, more advanced topics like tensorial fields in spacetime, exterior calculus and relativistic hydrodynamics are addressed. In the last, brief chapter the author gives a preview of gravity and shows where it becomes incompatible with Minkowsky spacetime. Well illustrated and enriched by many historical notes, this book also presents many applications of special relativity, ranging from particle physics (accelerators, particle collisions, quark-gluon plasma) to astrophysics (relativistic jets, active galactic nuclei), and including practical applications (Sagnac gyrometers, synchrotron radiation, GPS). In addition, the book provides some mathematical developments, such as the detailed analysis of the Lorentz group and its Lie algebra. The book is suitable for students in the third year of a physics degree or on a masters course, as well as researchers and any reader interested in relativity. Thanks to the geometric approach adopted, this book should also be beneficial for the study of general relativity. “A modern presentation of special relativity must put forward its essential structures, before illustrating them using concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Éric Gourgoulhon.” (excerpt from the Foreword by Thibault Damour)Table of ContentsMinkowski Spacetime.- Worldlines and Proper Time.- Observers.- Kinematics 1: Motion with Respect to an Observer.- Kinematics 2: Change of Observer.- Lorentz Group.- Lorentz Group as a Lie Group.- Inertial Observers and Poincaré Group.- Energy and Momentum.- Angular Momentum.- Principle of Least Action.- Accelerated Observers.- Rotating Observers.- Tensors and Alternate Forms.- Fields on Spacetime.- Integration in Spacetime.- Electromagnetic Field.- Maxwell Equations.- Energy-Momentum Tensor.- Energy-Momentum of the Electromagnetic Field.- Relativistic Hydrodynamics.- What about Relativistic Gravitation?.- A Basic Algebra.- B Web Pages.- C Special Relativity Books.

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Book SynopsisNew concise edition with a new introduction, abridged for the modern reader. The Principia. Mathematical Principles of Natural Philosophy is one of the most important scientific works ever to have been written and has had a profound impact on modern science. Consisting of three separate books, the Principia states Newton’s laws of motion and Newton’s law of universal gravitation. Understanding and acceptance of these theories was not immediate, however by the end of the seventeenth century no one could deny that Newton had far exceeded all previous works and revolutionised scientific thinking. The FLAME TREE Foundations series features core publications which together have shaped the cultural landscape of the modern world, with cutting-edge research distilled into pocket guides designed to be both accessible and informative.

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Book SynopsisThis textbook offers a detailed and self-contained presentation of quantum field theory, suitable for advanced undergraduate and graduate level courses. The author provides full derivations wherever possible and adopts a pedagogical tone without sacrificing rigour. A fully worked solutions manual is available online for instructors.Trade Review'This new and very welcome introduction to quantum field theory takes the reader from the basics of classical physics and the beauty of group theory to the intricacies and elegance of gauge field theories. Students and researchers alike will treasure this fresh approach to one of the foundation stones of modern physics.' Thomas Appelquist, Yale University'I wish this text had been available the last time I taught quantum field theory. The author provides clear, detailed expositions, which serve students with diverse backgrounds for multiple course syllabi.' Steve Gottlieb, Indiana University'The rigorous and logical approach makes this text certainly one to be seriously considered for use in a quantum field theory course. In any case, it is one which practitioners will definitely want to have within easy reach on their bookshelf.' Barry Holstein, University of Massachusetts Amherst'Both as an introductory text and as an excellent single-volume compendium on quantum field theory, this book is highly recommended for students as well as practitioners at all levels.' Wolfram Weise, Technical University of MunichTable of Contents1. Lorentz and Poincare Invariance; 2. Classical Mechanics; 3. Relativistic Classical Fields; 4. Relativistic Quantum Mechanics; 5. Introduction to Particle Physics; 6. Formulation of Quantum Field Theory; 7. Interacting Quantum Field Theories; 8. Symmetries and Renormalization; 9. Nonabelian Gauge Theories.

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Book SynopsisThis book and CD-ROM package has been developed in a lab-oriented course taught at Cal Tech in 1995 and 1996, and simultaneously augmented by artificial life research conducted there.Table of Contents1 Flavors of Artificial Life.- 1.1 Whither a Theory of the Living State?.- 1.2 Emulation and Simulation.- 1.3 Carbon-Based Artificial Life.- 1.4 Turing and von Neumann Automata.- 1.5 Cellular Automata.- 1.6 Overview.- 2 Artificial Chemistry and Self-Replicating Code.- 2.1 Virtual Machines and Self-Reproducing CA.- 2.2 Viruses and Core Worlds.- 2.3 The tierra System.- 2.4 avida, amoeba, and the Origin of Life.- 2.5 Overview.- 3 Introduction to Information Theory.- 3.1 Information Theory and Life.- 3.2 Channels and Coding.- 3.3 Uncertainty and Shannon Entropy.- 3.4 Joint and Conditional Uncertainty.- 3.5 Information.- 3.6 Noiseless Coding.- 3.7 Channel Capacity and Fundamental Theorem.- 3.8 Information Transmission Capacity for Genomes.- 3.9 Overview.- 4 Statistical Mechanics and Thermodynamics.- 4.1 Phase Space and Statistical Distribution Function.- 4.2 Averages, Ergodicity, and the Ergodic Theorem.- 4.3 Thermodynamical Equilibrium, Relaxation.- 4.4 Energy.- 4.5 Entropy.- 4.6 Second Law of Thermodynamics.- 4.7 Tèmperature.- 4.8 The Gibbs Distribution.- 4.9 Nonequilibrium Thermodynamics.- 4.10 First-Order Phase Transitions.- 4.11 Overview.- 5 Complexity of Simple Living Systems.- 5.1 Complexity and Information.- 5.2 The Maxwell Demon.- 5.3 Kolmogorov Complexity.- 5.4 Physical Complexity and the Natural Maxwell Demon.- 5.5 Complexity of tRNA.- 5.6 Complexity in Artificial Life.- 5.7 Overview.- 6 Self-Organization to Criticality.- 6.1 Self-Organization and Sandpiles.- 6.2 SOC in Forest Fires.- 6.3 SOC in the Living State.- 6.4 Theories of SOC.- 6.5 Overview.- 7 Percolation.- 7.1 Site Percolation.- 7.2 Cluster Size Distribution.- 7.3 Percolation in 1D.- 7.4 Higher-Dimensional Euclidean Lattices.- 7.5 Percolation on the Bethe Lattice.- 7.6 Scaling Theory.- 7.7 Percolation and Evolution.- 7.8 Overview.- 8 Fitness Landscapes.- 8.1 Theoretical Formulation.- 8.2 Example Landscapes.- 8.3 Fractal Landscapes.- 8.4 Diffusive and Nondiffusive Processes.- 8.5 RNA Landscapes.- 8.6 Fitness Landscape in avida.- 8.7 Overview.- 9 Experiments with avida.- 9.1 Choice of Chemistry.- 9.2 A Simple Experiment.- 9.3 Experiments in Adaptation.- 9.4 Experiments with Species and Genetic Distance.- 9.5 Overview.- 10 Propagation of Information.- 10.1 Information Transport and Equilibrium.- 10.2 The Artificial Life System sanda.- 10.3 Diffusion and Waves.- 10.4 Comparison: Theory and Experiment.- 10.5 Overview.- 11 Adaptive Learning at the Error Threshold.- 11.1 Information Processing at the Edge of Chaos.- 11.2 Adaptation to Computation in avida.- 11.3 Eigen’s Error Threshold.- 11.4 Molecular Evolution as an Ising Model.- 11.5 The Race to the Error Threshold.- 11.6 Approach to Error Threshold in avida.- 11.7 Overview.- A The avida User’s Manual.- A.1 Introduction.- A.2 A Beginner’s Guide to avida.- A.3 Time Slicing and the Fitness Landscape.- A.4 Reproduction.- A.5 The Virtual Computer.- A.6 Mutations.- A.7 Installing avida.- A.8 The Text Interface.- A.9 Configuring avida Runs.- A.10 Guide to Output Files.- A.11 Summary of Variables.- A.12 Glossary.- References.

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Book SynopsisTable of Contents1. The Statistical Basis of Thermodynamics 2. Elements of Ensemble Theory 3. The Canonical Ensemble 4. The Grand Canonical Ensemble 5. Formulation of Quantum Statistics 6. The Theory of Simple Gases 7. Ideal Bose Systems 8. Ideal Fermi Systems 9. Thermodynamics of the Early Universe 10. Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions 11. Statistical Mechanics of Interacting Systems: The Method of Quantized Fields 12. Phase Transitions: Criticality, Universality, and Scaling 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 14. Phase Transitions: The Renormalization Group Approach 15. Fluctuations and Nonequilibrium Statistical Mechanics 16. Computer Simulations

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Book SynopsisTable of Contents1. Introduction 2. The Basis of Monte Carlo 3. Pseudorandom Number Generators 4. Sampling, Scoring, and Precision 5. Variance Reduction Techniques 6. Markov Chain Monte Carlo 7. Inverse Monte Carlo 8. Linear Operator Equations 9. The Fundamentals of Neutral Particle Transport 10. Monte Carlo Simulation of Neutral Particle Transport 11. Monte Carlo Applications

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Book SynopsisThe latest volume in The New York Times bestselling physics series explains Einstein''s masterpiece: the general theory of relativityHe taught us classical mechanics, quantum mechanics and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein''s general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler).An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe''s real structure.

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Book SynopsisA distinctive and accessible introduction to quantum information science and quantum computing, this textbook provides a solid conceptual and formal understanding of quantum states and entanglement for undergraduate students and upper-level secondary school students with little or no background in physics, computer science, or mathematics.Trade ReviewWhile broadly accessible, the textbook does not dodge providing a solid conceptual and formal understanding of quantum states and entanglement - the key ingredients in quantum computing. The authors dish up a hearty meal for the readers, disentangling and explaining many of the classic quantum algorithms that demonstrate how and when QC has an advantage over classical computers. The book is spiced with Try Its, brief exercises that engage the readers in problem solving (both with and without mathematics) and help them digest the many counter-intuitive quantum information science and quantum computing concepts. * zb Math Open *This is a refreshing, pedagogical, and timely overview of quantum computing for non-experts, by two well-qualified authors. * Shimon Kolkowitz, University of Wisconsin-Madison *This is a much needed bridge between popular and technical texts that provides easy access to the topic of quantum computing for curious readers who aim to go further and deeper in their understanding. * Dieter Jaksch, University of Oxford *The reader gets to avoid the complexity of technical quantum-computing books, yet gets more depth and rigor than in the popular writing on the topic...the book is written in a very conversational rather than academic tone. * Bogdan Hoanca, University of Alaska Anchorage *Table of Contents1: Introduction 2: Traditional Computing 3: Traditional Bits in New Clothes 4: Qubits and Quantum States 5: Quantum Measurements 6: Quantum Gates 7: Putting a Spin on Spin 8: My Basis, Your Basis 9: Multi-qubit Systems, Entanglement, and Quantum Weirdness 10: Quantum Circuits and Multi-qubit Applications 11: Quantum Computing Algorithms 12: More Quantum Algorithms 13: RSA Encryption and the Shor Factoring Algorithm 14: Fundamental Quantum Issues 15: Complexifying Quantum States 16: Present and Future QIS and QC

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Book SynopsisQuantum Computing: From Alice to Bob provides a distinctive and accessible introduction to the rapidly growing fields of quantum information science and quantum computing. The textbook is designed for undergraduate students and upper-level secondary school students with little or no background in physics, computer science, or mathematics beyond secondary school algebra and a bit of trigonometry. Higher education faculty members and secondary school mathematics, physics, and computer science educators who want to learn about quantum computing and perhaps teach a course accessible to students with wide ranging backgrounds will also find the book useful and enjoyable. While broadly accessible, the textbook does not dodge providing a solid conceptual and formal understanding of quantum states and entanglement - the key ingredients in quantum computing. The authors dish up a hearty meal for the readers, disentangling and explaining many of the classic quantum algorithms that demonstrate how and when QC has an advantage over classical computers. The book is spiced with Try Its, brief exercises that engage the readers in problem solving (both with and without mathematics) and help them digest the many counter-intuitive quantum information science and quantum computing concepts.Trade ReviewWhile broadly accessible, the textbook does not dodge providing a solid conceptual and formal understanding of quantum states and entanglement - the key ingredients in quantum computing. The authors dish up a hearty meal for the readers, disentangling and explaining many of the classic quantum algorithms that demonstrate how and when QC has an advantage over classical computers. The book is spiced with Try Its, brief exercises that engage the readers in problem solving (both with and without mathematics) and help them digest the many counter-intuitive quantum information science and quantum computing concepts. * zb Math Open *This is a refreshing, pedagogical, and timely overview of quantum computing for non-experts, by two well-qualified authors. * Shimon Kolkowitz, University of Wisconsin-Madison *This is a much needed bridge between popular and technical texts that provides easy access to the topic of quantum computing for curious readers who aim to go further and deeper in their understanding. * Dieter Jaksch, University of Oxford *The reader gets to avoid the complexity of technical quantum-computing books, yet gets more depth and rigor than in the popular writing on the topic...the book is written in a very conversational rather than academic tone. * Bogdan Hoanca, University of Alaska Anchorage *Table of Contents1: Introduction 2: Traditional Computing 3: Traditional Bits in New Clothes 4: Qubits and Quantum States 5: Quantum Measurements 6: Quantum Gates 7: Putting a Spin on Spin 8: My Basis, Your Basis 9: Multi-qubit Systems, Entanglement, and Quantum Weirdness 10: Quantum Circuits and Multi-qubit Applications 11: Quantum Computing Algorithms 12: More Quantum Algorithms 13: RSA Encryption and the Shor Factoring Algorithm 14: Fundamental Quantum Issues 15: Complexifying Quantum States 16: Present and Future QIS and QC

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Book SynopsisThis textbook uses insight from differential equations to analyse fundamental subjects of modern theoretical physics, including classical and quantum mechanics, thermodynamics, electromagnetism, superconductivity, gravitational physics, and quantum field theories.Table of ContentsPreface Notation and Convention 1: Hamiltonian Systems and Applications 2: Schrödinger Equation and Quantum Mechanics 3: Maxwell Equations, Dirac Monopole, and Gauge Fields 4: Special Relativity 5: Abelian Gauge Field Equations 6: Dirac Equations 7: GinzburgDSLandau Equations for Superconductivity 8: Magnetic Vortices in Abelian Higgs Theory 9: Non-Abelian Gauge Field Equations 10: Einstein Equations and Related Topics 11: Charged Vortices and ChernDSSimons Equations 12: Skyrme Model and Related Topics 13: Strings and Branes 14: BornDSInfeld Theory of Electromagnetism 15: Canonical Quantization of Fields Appendices Bibliography Index

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Book SynopsisThis textbook uses insight from differential equations to analyse fundamental subjects of modern theoretical physics, including classical and quantum mechanics, thermodynamics, electromagnetism, superconductivity, gravitational physics, and quantum field theories.Table of ContentsPreface Notation and Convention 1: Hamiltonian Systems and Applications 2: Schrödinger Equation and Quantum Mechanics 3: Maxwell Equations, Dirac Monopole, and Gauge Fields 4: Special Relativity 5: Abelian Gauge Field Equations 6: Dirac Equations 7: GinzburgDSLandau Equations for Superconductivity 8: Magnetic Vortices in Abelian Higgs Theory 9: Non-Abelian Gauge Field Equations 10: Einstein Equations and Related Topics 11: Charged Vortices and ChernDSSimons Equations 12: Skyrme Model and Related Topics 13: Strings and Branes 14: BornDSInfeld Theory of Electromagnetism 15: Canonical Quantization of Fields Appendices Bibliography Index

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Book SynopsisThis book presents the reader with modern tools, approaches, approximations, and applications of quantum mechanics. Quantum mechanics forms the foundation of all modern physics, including atomic, nuclear, and molecular physics, the physics of the elementary particles, condensed matter physics, and also modern astrophysics.Trade Review"The book teaches students how to approach and solve the types of quantum mechanical problems they will encounter throughout their careers. It will serve as an excellent text for a graduate level course." * C. Stephen Hellberg, Naval Research Laboratory *Table of Contents1. Schrodinger equation on a lattice ; 2. Dirac notation ; 3. Back to Schrodinger equation on the lattice ; 4. Operator-mechanics ; 5. Time evolution and wave packets ; 6. Simulaneaous observables ; 7. Continuity equation and wavefunction properties ; 8. Bond states in one-dimension ; 9. Scattering in one dimension ; 10. Periodic Potentials ; 11. The harmonic oscillator ; 12. WKB approximation ; 13. Quantum mechanics and path integrals ; 14. Applications of path integrals ; 15. Angular momentum ; 16. Bound states in spherically symmetric potentials ; 17. The hydrogen-like atom ; 18. Angular momentum and spherical symmetry ; 19. Scattering in 3D ; 20. Time independent perturbation expansion ; 21. Applications of perturbation theory ; 22. Time-dependent Hamiltonian ; 23. Spin angular momentum ; 24. Adding angular momenta ; 25. Identical particles ; 26. Elementary atomic physics ; 27. Molecules ; 28. The elasticity field ; 29. Quantization of the free electromagnetic field ; 30. Interaction of radiation with charged particles ; 31. Elementary relativistic quantum mechanics

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Book SynopsisThe book introduces classical mechanics. It does so in an informal style with numerous fresh, modern and inter-disciplinary applications assuming no prior knowledge of the necessary mathematics. The book provides a comprehensive and self-contained treatment of the subject matter up to the forefront of research in multiple areas.Table of ContentsPart I: Newtonian Mechanics 1: Introduction 2: Newton's Three Laws 3: Energy and Work 4: Introductory Rotational Dynamics 5: The Harmonic Oscillator 6: Wave Mechanics & Elements of Mathematical Physics Part II: Langrangian Mechanics 7: Introduction 8: Coordinates & Constraints 9: The Stationary Action Principle 10: Constrained Langrangian Mechanics 11: Point Transformations in Langrangian Mechanics 12: The Jacobi Energy Function 13: Symmetries & Langrangian-Hamiltonian-Jacobi Theory 14: Near-Equilibrium Oscillations 15: Virtual Work & d'Alembert's Principle Part III: Canonical Mechanics 16: Introduction 17: The Hamiltonian & Phase Space 18: Hamiltonian's equations & Routhian Reduction 19: Poisson Brackets & Angular momentum 20: Canonical & Gauge Transformations 21: Hamilton-Jacobi Theory 22: Liouville's Theorem & Classical Statistical Mechanics 23: Constrained Hamiltonian Dynamics 24: Autonomous Geometrical Mehcanics 25: The Structure of Phase Space 26: Near-Integrable Systems Part IV: Classical Field Theory 27: Introduction 28: Langrangian Field Theory 29: Hamiltonian Field Theory 30: Clssical Electromagnetism 31: Neother's Theorem for Fields 32: Classical Path-Integrals Part V: Preliminary Mathematics 33: The (Not so?) Basics 34: Matrices 35: Partial Differentiation 36: Legendre Transformations 37: Vector Calculus 38: Differential equations 39: Calculus of Variations Part VI: Advanced Mathematics 40: Linear Algebra 41: Differential Geometry Part VII: Exam Style Questions Appendix A: Noether's Theorem Explored Appendix B: The Action Principle Explored Appendix C: Useful Relations Appendxi D: Poisson & Nambu Brackets Explored Appendix: Canonical Transformations Explored Appendix F: Action-Angle Variables Explored Appendix G: Statistical Mechanics Explored Appendix H: Biographies

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Book SynopsisBlack holes present one of the most fascinating predictions of Einstein''s general theory of relativity. There is strong evidence of their existence through observation of active galactic nuclei, including the centre of our galaxy, observations of gravitational waves, and others.There exists a large scientific literature on black holes, including many excellent textbooks at various levels. However, most of these steer clear from the mathematical niceties needed to make the theory of black holes a mathematical theory. Those which maintain a high mathematical standard are either focused on specific topics, or skip many details. The objective of this book is to fill this gap and present a detailed, mathematically oriented, extended introduction to the subject.The book provides a wide background to the current research on all mathematical aspects of the geometry of black hole spacetimes.Trade ReviewWritten with a high standard of rigor and care, with very good treatments of many topics that are hard to find elsewhere. * Robert Wald, University of Chicago *Including some very interesting and unique material, the book is written in a manner that will be accessible for students, and provide a valuable resource for experts working in mathematical general relativity. * Greg Galloway, University of Miami *This text is an excellent research level monograph exploring the detailed and rich structure of black holes in mathematical physics. * Kymani Armstrong-Williams, Physics Book Reviews *Table of ContentsPART I GLOBAL LORENTZIAN GEOMETRY 1: Basic Notions 2: Elements of causality 3: Some applications PART II BLACK HOLES 4: An introduction to black holes 5: Further selected solutions 6: Extensions, conformal diagrams 7: Projection diagrams 8: Dynamical black holes

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Book SynopsisQuantum theory (QT) is the best, most useful physics theory ever invented. For example, ubiquitous are cell phones, laser scanners, medical imagers, all inventions depending on QT. However, there is something deeply wrong with QT. It describes the probabilities of what happens, but it does not give a description of what actually happens. Most (but not all) physicists are not worried about this flaw, the probabilities are good enough for them. Other physicists, the author included, believe that is not good enough. The purpose of physics is to describe reality. To not do so is to abandon ''the great enterprise'' (John Bell). This book shows one way to alter QT so that the new theory does describe what actually happens. This theory, created over three decades ago, has been called the ''Continuous Spontaneous Localization'' (CSL) theory.Many experiments over this period have tested CSL, and so far it is neither confirmed nor refuted. This book shows how CSL works, and discusses its consequTrade ReviewA most welcome addition to the physics literature written with extreme care and covering the objective subject matter in a thorough professional and methodical manner. * Daniel Sudarsky, UNAM, Mexico City *A book of very high quality presenting a way of modifying quantum mechanics to remove some of its most serious problems (especially the measurement problem). * Kelvin McQueen, Chapman University, Orange, California *Pearle is the master of this material and writes with beautiful clarity and well-judged occasional witticisms and side-remarks. His experience as teacher, as well as researcher, shows in the vivid explanations, and the careful and consistent level of detail in the exposition. * Jeremy Butterfield, University of Cambridge *Table of Contents1: Introduction 2: Continuous Spontaneous Localization (CSL) Theory 3: CSL Theory Refinements 4: Non-Relativistic CSL 5: Spontaneous Localization (SL) Theory 6: Some Experiments Testing CSL 7: Interpretational Remarks 8: Supplement to Chapter 1 9: Supplement to Chapter 2 10: Supplement to Chapter 3 11: Supplement to Chapter 4 12: Supplement to Chapter 5 13: Supplement to Chapter 6 14: Supplement to Chapter 7 15: A Stochastic Differential Equation Cookbook 16: CSL Expressed as a Schrodinger Stochastic DE 17: Applying the CSL Stratonovich Equation to the Free Particle Undergoing Collapse in Position 18: Applying the CSL Stratonovich Equation to the Harmonic Oscillator Undergoing Collapse in Position Appendix A: Gaussians Appendix B: Random Walk Appendix C: Brownian Motion/Wiener Process Appendix D: White Noise Appendix E: White Noise Field Appendix F: Density Matrix Appendix G: Theoretical Constraint Calculations

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Book SynopsisA series of seminal technological revolutions has led to a new generation of electronic devices miniaturized to such tiny scales where the strange laws of quantum physics come into play. There is no doubt that, unlike scientists and engineers of the past, technology leaders of the future will have to rely on quantum mechanics in their everyday work. This makes teaching and learning the subject of paramount importance for further progress. Mastering quantum physics is a very non-trivial task and its deep understanding can only be achieved through working out real-life problems and examples. It is notoriously difficult to come up with new quantum-mechanical problems that would be solvable with a pencil and paper, and within a finite amount of time. This book remarkably presents some 700+ original problems in quantum mechanics together with detailed solutions covering nearly 1000 pages on all aspects of quantum science. The material is largely new to the English-speaking audience. The proTrade ReviewIn his Preface, Victor Galitski, Jr. offers something of an apology for preserving an old-school style to the contents. Nice as it is no such apology is called for with such an excellent book. The publisher, OUP, is to be congratulated on the investment of a professional indexer, who has done a good job. * S.W. Lovesey, Contemporary Physics, *An excellent resource for students and teachers seeking a deep understanding of quantum mechanics * Dr David Bowler, UCL *Finally, the reader receives the English translation of this magnificent book, arguably, the best collection of working problems in Quantum Mechanics. My congratulations are going to thousands of students and working physicists who will definitely find here the material for exercises as well as an inspiration in original research. * David Khmelnitskii, Cavendish Laboratory, Cambridge *Most physicists and physics students will affirm that they learned the subject by working the problems. Here is a treasure trove of quantum problems and solutions - a splendid resource for teachers trying to expand the repertoire of their problem sets and for students of all ages trying to deepen their understanding of the heart of modern physics. * William D. Phillips, NIST, Nobel Laureate Physics 1997 *Provides a wide range of opportunities to learn what quantum mechanics does through an impressive collection of solved problems. [...] The result is a gem of old-world craftsmanship, well worth a place alongside the other classic texts of quantum mechanics in any physicist's library. * Physics Today, *This is a must-have book for anybody who wants to gain working knowledge of quantum mechanics. It gives both fundamental physical understanding and concrete knowledge of specific technical methods and approaches. * Eugene Demler, Harvard University *A treasure-trove of insightful problems and solutions, 'Exploring Quantum Mechanics' provides a unique and rare perspective on quantum physics. Spanning a broad range of subfields, it is a testament to the mastery of the original authors, Galitski Sr. et al., and the translator, Galitski Jr. Students and specialists of quantum mechanics in the English speaking science world will greatly benefit from this invaluable collection. * Gil Refael, CalTech *This collection of problems in quantum physics, probably the largest of its kind in the world, gives the reader the unique possibility to learn to feel at home in the world of quantum mechanics. It includes more than seven hundred problems of various difficulty accompanied by detailed solutions, ranging from elementary single-particle quantum mechanics in one dimension to relativistic field theory and advanced aspects of nuclear physics. * Andrey Varlamov, Italian National Research Council *Table of ContentsPreface ; 1. Operators in Quantum Mechanics ; 2. One-Dimensional Motion ; 3. Orbital Angular Momentum ; 4. Motion in a Spherically-symmetric Potential ; 5. Spin ; 6. Time-Dependent Quantum Mechanics ; 7. Motion in a Magnetic Field ; 8. Perturbation Theory; Variational Method; Sudden and Adiabatic Theory ; 9. Quasi-Classical Approximation; 1/N-Expansion in Quantum Mechanics ; 10. Identical particles; Second quantization ; 11. Atoms and Molecules ; 12. Atomic Nucleus ; 13. Particle Collisions ; 14. Quantum Radiation Theory ; 15. Relativistic Wave Equations ; 16. Appendix

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Book SynopsisThis short guide to modern error analysis is primarily intended to be used in undergraduate laboratories in the physical sciences. No prior knowledge of statistics is assumed. The necessary concepts are introduced where needed and illustrated graphically. The book emphasises the use of computers for error calculations and data fitting.Trade ReviewWith the shift from analytic methods to spreadsheet-based techniques, this book will enable students simultaneously to (a) become fluent in the choice and application of appropriate methods (b) understand the underlying principles. * David Saxon, University of Glasgow *This is a rather beautiful little book. * David J. Hand, International Statistical Review *Table of Contents1. Errors in the physical sciences ; 2. Random errors in measurement ; 3. Uncertainties as probabilities ; 4. Error propagation ; 5. Data visualisation and reduction ; 6. Least-squares fitting of complex functions ; 7. Computer minimisation and the error matrix ; 8. Hypothesis testing - how good are our models ; 9. Topics for further summary

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Book SynopsisThis book collects lecture courses and seminars given at the Les Houches Summer School 2010 on "Quantum Theory: From Small to Large Scales". It reviews the state-of-the-art developments in this field by touching on different research topics from an interdisciplinary perspective.Table of Contents1. A Word from the Organizers ; 2. Mass Renormalization in Nonrelativstic Quantum Electrodynamics ; 3. Quantum Brownian Motion ; 4. The Temporal Ultraviolet Limit ; 5. Locality in Quantum Systems ; 6. Entropic Fluctuations in Quantum Statistical Mechanics - An Introduction ; 7. Quantum Phase Transitions: Introduction and Some Open Problems ; 8. Cold Quantum Gases and Bose-Einstein Condensation ; 9. Renormalization Group and Problem of Radiation ; 10. SUSY Statistical Mechanics and Random Band Matrices ; 11. Long-time diffusion for a quantum particle ; 12. Universality of Generalized Wigner Matrices ; 13. On Transport in Quantum Devices ; 14. The ground state construction of the two-dimennsion Hubbard model on the honeycomb lattice

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Book SynopsisBrownian diffusion is the motion of one or more solute molecules in a sea of very many, much smaller solvent molecules. Its importance today owes mainly to cellular chemistry, since Brownian diffusion is one of the ways in which key reactant molecules move about inside a living cell. This book focuses on the four simplest models of Brownian diffusion: the classical Fickian model, the Einstein model, the discrete-stochastic (cell-jumping) model, and the Langevin model. The authors carefully develop the theories underlying these models, assess their relative advantages, and clarify their conditions of applicability. Special attention is given to the stochastic simulation of diffusion, and to showing how simulation can complement theory and experiment. Two self-contained tutorial chapters, one on the mathematics of random variables and the other on the mathematics of continuous Markov processes (stochastic differential equations), make the book accessible to researchers from a broad spectrum of technical backgrounds.Trade ReviewIn a lively tutorial style, the authors discuss some of the most widely used mathematical formulations of diffusion. They have endeavored to organize and present the subject matter from a purely logical perspective. They emphasize the basic physical assumptions and the conditions for the validity of each of the mathematical formalisms. No subtlety is bypassed, and no limitation of the theory is swept under the carpet. * Debashish Chowdhury, Physics today *In a lively tutorial style, the authors discuss some of the most widely used mathematical formulations of diffusion. They have endeavored to organize and present the subject matter "from a purely logical perspective". They emphasize the basic physical assumptions and the conditions for the validity of each of the mathematical formalisms. No subtlety is bypassed, and no limitation of the theory is swept under the carpet. * Physics Today *Table of Contents1. The Fickian theory of diffusion ; 2. A review of random variable theory ; 3. Einstein's theory of diffusion ; 4. Implications and limitations of the Einstein theory of diffusion ; 5. The discrete-stochastic approach ; 6. Master equations and simulation algorithms for the discrete-stochastic approach ; 7. Continuous Markov process theory ; 8. Langevin's theory of diffusion ; 9. Implications of Langevin's theory ; 10. Diffusion in an external force field ; 11. The first-passage time approach

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Book SynopsisThe Physics of Quantum Mechanics aims to give students a good understanding of how quantum mechanics describes the material world. The text stresses the continuity between the quantum world and the classical world, which is merely an approximation to the quantum world.Trade ReviewThis book is a deep, well-explained and beautiful text on the foundations and applications of quantum mechanics. It is eminently suitable for advanced undergraduates and graduates who wish to study the subject. Some precious jewels can be found within after building up the Dirac representation of quantum mechanics: scattering theory and condensed matter applications, for example. * Ben Allanach, Department of Applied Mathematics and Theoretical Physics, University of Cambridge *The extensive discussion of the physics behind the mathematical manipulations of the theory, coupled with the smooth, colloquial writing style and delightful historical footnotes makes this book somewhat unique in the field. It devotes large sections to the more modern topics of quantum computing and quantum measurement theory, which are active areas of current research. In addition, there is a copious selection of problems, at all levels of difficulty, which should prove extremely useful to anyone teaching the course. * Harold S. Zapolsky, Rutgers University *Binney and Skinners introductory book on quantum mechanics approaches the subject in a unique way ... The text is very well written for the target audience of second or third year University students in Physics, Chemistry, or certain Engineering specialties and I would highly recommend it for anyone who might be considering teaching or tutoring such a course. * Brian Todd Huffman, University of Oxford *Table of Contents1. Introduction ; 2. Operators, measurement and time evolution ; 3. Oscillators ; 4. Transformations & Observables ; 5. Motion in step potentials ; 6. Composite systems ; 7. Angular Momentum ; 8. Hydrogen ; 9. Motion in a magnetic field ; 10. Perturbation theory ; 11. Helium and the periodic table ; 12. Adiabatic principle ; 13. Scattering Theory ; Appendices

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Book SynopsisThe Nature of Complex Networks provides a systematic introduction to the statistical mechanics of complex networks and the different theoretical achievements in the field that are now finding strands in common.The book presents a wide range of networks and the processes taking place on them, including recently developed directions, methods, and techniques. It assumes a statistical mechanics view of random networks based on the concept of statistical ensembles but also features the approaches and methods of modern random graph theory and their overlaps with statistical physics.This book will appeal to graduate students and researchers in the fields of statistical physics, complex systems, graph theory, applied mathematics, and theoretical epidemiology.Trade ReviewThe current volume by Dorogovtsev and Mendes takes quite a broad view of complex networks to include the analysis of finite and infinite graphs, directed and undirected graphs, multigraphs, hypergraphs, and even simplicial complexes, as networks scale according to increasing N or in some other fashion. The writing style is that of physics and especially statistical mechanics with frequent connections made to physical concepts such as Bose-Einstein condensation...The current volume can especially serve as a useful reference on complex networks from a physics perspective. * Lenwood S. Heath, MathSciNet *Table of ContentsPreface 1: First insight 2: Graphs 3: Classical random graphs 4: Equilibrium networks 5: Evolving networks 6: Connected components 7: Epidemics and spreading phenomena 8: Networks of networks 9: Spectra and communities 10: Walks and search 11: Temporal networks 12: Cooperative systems on networks 13: Inference and reconstruction 14: What's next? Further Reading Appendices A-G References

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Book SynopsisComputation is the process of applying a procedure or algorithm to the solution of a mathematical problem. This book covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum mechanics in computation.Trade Review"A beautiful little book.... It succeeds so well because Geroch believes that 'physics is a human activity' and wants to share some of its joy with others." - Physics Today"

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Book SynopsisBased on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students.Trade Review`Shankar obviously enjoys his mathematics, and his attitude toward mathematics is simultaneously refreshing and contagious....Dirac notation is intriguingly introduced in the discussion of vector spaces. Finally, the book is richly endowed with well-chosen problems.' American Journal of Physics `Consistent with the needs of science students...a sound mathematical reference for anyone studying or practicing in the physical sciences.' Choice Table of ContentsDifferential Calculus of One Variable. Integral Calculus. Calculus of Many Variables. Infinite Series. Complex Numbers. Functions of a Complex Variable. Vector Calculus. Matrices and Determinants. Linear Vector Spaces. Differential Equations. Answers. Index.

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Book Synopsis1. Introductory Material.- 2. Green's Functions at Zero Temperature.- 3. Nonzero Temperatures.- 4. Exactly Solvable Models.- 5. Homogeneous Electron Gas.- 6. Strong Correlations.- 7. Electron-Phonon Interaction.- 8. dc Conductivities.- 9. Optical Properties of Solids.- 10. Superconductivity.- 11. Superfluids.- References.- Author Index.Trade ReviewAbout the First Edition: `This is a worthy newcomer and will make an excellent teaching text.' Physics Bulletin `The book should serve as a valuable contribution to the library of students and researchers in solid state and theoretical physics.' Philosophical Magazine About the Second Edition: `Since its first edition, this book has become one of the most popular textbooks in quantum many-body theory, thus guaranteeing the interest of the scientific community in this second edition.' Mathematical Reviews Table of Contents1: Introductory Material. 1.1. Harmonic Oscillators and Phonons. 1.2. Second Quantization for Particles. 1.3. Electron-Phonon Interactions. 1.4. Spin Hamiltonians. 1.5. Photons. 1.6. Pair Distribution Function. 2: Green's Functions at Zero Temperature. 2.1. Interaction Representation. 2.2. S Matrix. 2.3. Green's Functions. 2.4. Wick's Theorem. 2.5. Feynman Diagrams. 2.6. Vacuum Polarization Graphs. 2.7. Dyson's Equation. 2.8. Rules for Constructing Diagrams. 2.9. Time-Loop S Matrix. 2.10. Photon Green's Functions. 3: Nonzero Temperatures. 3.1. Introduction. 3.2. Matsubara Green's Functions. 3.3. Retarded and Advanced Green's Functions. 3.4. Dyson's Equation. 3.5. Frequency Summations. 3.6. Linked Cluster Expansions. 3.7. Real Time Green's Functions. 3.8. Kubo Formula for Electrical Conductivity. 3.9. Other Kubo Formulas. 4: Exactly Solvable Models. 4.1. Potential Scattering. 4.2. Localized State in the Continuum. 4.3. Independent Boson Models. 4.4. Bethe Lattice. 4.5. Tomonaga Model. 4.6. Polaritons. 5: Homogeneous Electron Gas. 5.1. Exchange and Correlation. 5.2. Wigner Lattice. 5.3. Metallic Hydrogen. 5.4. Linear Screening. 5.5. Model Dielectric Functions. 5.6. Properties of the Electron Gas. 5.7. Sum Rules. 5.8. One-Electron Properties. 6: Strong Correlations. 6.1. Kondo Model. 6.2. Single-Site Anderson Model. 6.3. Hubbard Model. 6.4. Hubbard Model: Magnetic Phases. 7: Electron-Phonon Interaction. 7.1. Fröhlich Hamiltonian. 7.2. Small Polaron Theory. 7.3. Heavily Doped Semiconductors. 7.4. Metals. 8: dc Conductivities. 8.1. Electron Scattering by Impurities. 8.2. Mobility of Frölich Polarons. 8.3. Electron-Phonon Relaxation Times. 8.4. Electron-Phonon Interactions in Metals. 8.5. Quantum Boltzmann Equation. 8.6. Quantum Dot Tunneling. 9: Optical Properties of Solids. 9.1. Nearly Free-Electron Systems. 9.2. Wannier Excitons. 9.3. X-Ray Spectra in Metals. 10: Superconductivity. 10.1. Cooper Instability. 10.2. Superconducting Tunneling. 10.3. Strong Coupling Theory. 10.4. Transition Temperature. 11: Superfluids. 11.1. Liquid 4He. 11.2. Liquid 3He. 11.3. Quantum Hall Effects.

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Book SynopsisTable of Contents1. Galilean relativity 2. Lorentz Boosts 3. Development of the Formalism 4. Electrodynamics 5. Gravity 6. Experiments and Applications Part II: Mathematics 7. Mathematics of Translations 8. The Rotation Group 9. The Lorentz Group

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Book SynopsisIntroduction: Newtonian Physics And Special Relativity.- Relativity Principles and Gravitation.- The Special Theory of Relativity.- The Mathematics Of The General Theory Of Relativity.- Vectors, Tensors, and Forms.- Basis Vector Fields and the Metric Tensor.- Non-inertial Reference Frames.- Differentiation, Connections, and Integration.- Curvature.- Einstein's Field Equations.- Einstein's Field Equations.- The Linear Field Approximation.- The Schwarzschild Solution and Black Holes.- Cosmology.- Homogeneous and Isotropic Universe Models.- Universe Models with Vacuum Energy.- Anisotropic and Inhomogeneous Universe Models.- Advanced Topics.- Covariant Decomposition, Singularities, and Canonical Cosmology.- Spatially Homogeneous Universe Models.- Israel's Formalism: The Metric Junction Method.- Brane-worlds.- Kaluza-Klein Theory.Trade ReviewFrom the reviews: "A sophisticated treatment of general relativity with a considerable number of applications to cosmology. … The book may be read in several different ways, depending on the interests of readers. A rich source of material; college libraries should have it on their shelves. Summing Up: Highly recommended. Upper-division undergraduates through professionals." (K. L. Schick, CHOICE, Vol. v4 (3), November, 2007) "This book is a carefully prepared overview about all essential aspects of relativity. Its 5 parts give the standard way to present the material … . Differently from other textbooks, the present authors emphasize much more concrete problems and examples … . The book contains a very large number of formulas, partially due to the fact, that many statements are given both in the old index-notation as well as in the modern index-free notation; I think this is helpful for the readers." (Hans-Jürgen Schmidt, Zentralblatt MATH, Vol. 1126 (3), 2008) "Even if you already have a sizable collection of books on general relativity, or if you are looking for a good modern book to teach a course from, the text by Grøn and Hervik is … a valuable addition to your collection." (David H. Delphenich, Mathematical Reviews, Issue 2008 i)Table of ContentsI. Introduction: Newtonian Physics and Special Relativity- 1. Relativity Principals and Gravitation 2. The Special Theory of Relativity II. The Mathematics of the General Theory of Relativity- 3. Vectors, Tensors, and Forms 4. Basis Vector Fields and Metric Tensor 5. Non-inertial Reference Frames 6. Differentiation, Connections and Integration 7. Curvature II. Einstein's Field Equations- 8. Einstein's Field Equations 9. The Linear Field Approximation 10. The Schwarzschild Solution and Black Holes IV. Cosmology- 11. Homogeneous and Isotropic Universe Models 12. Universe Models with Vacuum Energy 13. An Anisotropic Universe V. Advanced Topics- 14. Covariant decomposition, Singularities, and Canonical Cosmology 15. Homogeneous Spaces 16. Israel's Formalism: The metric junction method 17. Brane-worlds 18. Kaluza-Klein Theory VI. Appendices- A. Constrants of Nature B. Penrose diagrams C. Anti-de Sitter spacetime D. Suggested further reading

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Book SynopsisDifferentiable Dynamical Systems.- Notes.- References for Notes.- What Is Global Analysis?.- Stability and Genericity in Dynamical Systems.- Personal Perspectives on Mathematics and Mechanics.- Dynamics in General Equilibrium Theory.- Some Dynamical Questions in Mathematical Economics.- Review of Global Variational Analysis: Weier strass Integrals on a Riemannian Manifold.- Review of Catastrophe Theory: Selected Papers.- On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff.- Robert Edward Bowen (jointly with J. Feldman and M. Ratner).- On How I Got Started in Dynamical Systems.Table of ContentsDifferentiable Dynamical Systems.- Notes.- References for Notes.- What Is Global Analysis?.- Stability and Genericity in Dynamical Systems.- Personal Perspectives on Mathematics and Mechanics.- Dynamics in General Equilibrium Theory.- Some Dynamical Questions in Mathematical Economics.- Review of Global Variational Analysis: Weier strass Integrals on a Riemannian Manifold.- Review of Catastrophe Theory: Selected Papers.- On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff.- Robert Edward Bowen (jointly with J. Feldman and M. Ratner).- On How I Got Started in Dynamical Systems.

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Book SynopsisThe Second Law of Thermodynamics has been called the most important law of nature: It is the law that gives a direction to processes that is not inherent in the laws of motion, that says the state of the universe is driven to thermal equilibrium.

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Book SynopsisIntended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics.Table of Contents1. Introduction; 2. Quantum Field Theory; 3. Statistical Mechanics; 4. Global Conformal Invariance; 5. Conformal Invariance in Two Dimensions; 6. The Operator Formalism I; 7. The Operator Formalism II; 8. Minimal Models; 9. The Coulomb Gas Formalism; 10. Modular Invariance; 11. Finite Size Scaling; 12. The Two-Dimensional Ising Model; 13. Simple Lie Algebras; 14. Affine Lie Algebras; 15. The WZNW Model; 16. Fusion Rules; 17. Modular Invariants; 18. The Coset Construction

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Book SynopsisRevised and updated, it includes expanded discussions of * the fundamentals of geostrophic turbulence * the theory of wave-mean flow interaction * thermocline theory * finite amplitude barocline instability.Trade ReviewFrom the reviews"The author has done a masterful job in presenting the theory with the necessary mathematical foundation, while keeping the physical aspects in clear view ... it is an outstanding introduction to a complex and important subject." (GEOPHYSICS)Table of ContentsPreliminaries * Fundamentals * Inviscid Shallow-Water Theory * Friction and Viscous Flow * Homogeneous Models of the Wind-Driven Oceanic Circulation * Quasigeostrophic Motion of a Stratified Fluid on a Sphere * Instability Theory * Ageostrophic Motion

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Book SynopsisI Newtonian Mechanics.- 1 Experimental facts.- 2 Investigation of the equations of motion.- II Lagrangian Mechanics.- 3 Variational principles.- 4 Lagrangian mechanics on manifolds.- 5 Oscillations.- 6 Rigid bodies.- III Hamiltonian Mechanics.- 7 Differential forms.- 8 Symplectic manifolds.- 9 Canonical formalism.- 10 Introduction to perturbation theory.- Appendix 1 Riemannian curvature.- Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids.- Appendix 3 Symplectic structures on algebraic manifolds.- Appendix 4 Contact structures.- Appendix 5 Dynamical systems with symmetries.- Appendix 6 Normal forms of quadratic hamiltonians.- Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories.- Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem.- Appendix 9 Poincaré's geometric theorem, its generalizations and applications.- Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters.- Appendix 11 Short wave asymptotics.- Appendix 12 Lagrangian singularities.- Appendix 13 The Korteweg-de Vries equation.- Appendix 14 Poisson structures.- Appendix 15 On elliptic coordinates.- Appendix 16 Singularities of ray systems.Trade ReviewSecond Edition V.I. Arnol’d Mathematical Methods of Classical Mechanics "The book's goal is to provide an overview, pointing out highlights and unsolved problems, and putting individual results into a coherent context. It is full of historical nuggets, many of them surprising . . . The examples are especially helpful; if a particular topic seems difficult, a later example frequently tames it. The writing is refreshingly direct, never degenerating into a vocabulary lesson for its own sake. The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview." —AMERICAN MATHEMATICAL MONTHLYTable of ContentsI Newtonian Mechanics.- 1 Experimental facts.- 2 Investigation of the equations of motion.- II Lagrangian Mechanics.- 3 Variational principles.- 4 Lagrangian mechanics on manifolds.- 5 Oscillations.- 6 Rigid bodies.- III Hamiltonian Mechanics.- 7 Differential forms.- 8 Symplectic manifolds.- 9 Canonical formalism.- 10 Introduction to perturbation theory.- Appendix 1 Riemannian curvature.- Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids.- Appendix 3 Symplectic structures on algebraic manifolds.- Appendix 4 Contact structures.- Appendix 5 Dynamical systems with symmetries.- Appendix 6 Normal forms of quadratic hamiltonians.- Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories.- Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov’s theorem.- Appendix 9 Poincaré’s geometric theorem, its generalizations and applications.- Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters.- Appendix 11 Short wave asymptotics.- Appendix 12 Lagrangian singularities.- Appendix 13 The Korteweg-de Vries equation.- Appendix 14 Poisson structures.- Appendix 15 On elliptic coordinates.- Appendix 16 Singularities of ray systems.

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Book SynopsisI Asymptotic Solutions of Evolution Problems.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VIII Bifurcation of Periodic Solutions in the General Case.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case.- XII Stability and Bifurcation in Conservative Systems.Table of ContentsI Asymptotic Solutions of Evolution Problems.- I.1 One-Dimensional, Two-Dimensional n-Dimensional, and Infinite-Dimensional Interpretations of (I.1).- I.2 Forced Solutions; Steady Forcing and T-Periodic Forcing; Autonomous and Nonautonomous Problems.- I.3 Reduction to Local Form.- I.4 Asymptotic Solutions.- I.5 Asymptotic Solutions and Bifurcating Solutions.- I.6 Bifurcating Solutions and the Linear Theory of Stability.- I.7 Notation for the Functional Expansion of F(t µ,U).- Notes.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- II.1 The Implicit Function Theorem.- II.2 Classification of Points on Solution Curves.- 1I.3 The Characteristic Quadratic. Double Points, Cusp Points, and Conjugate Points.- II.4 Double-Point Bifurcation and the Implicit Function Theorem.- II.5 Cusp-Point Bifurcation.- II.6 Triple-Point Bifurcation.- II.7 Conditional Stability Theorem.- II.8 The Factorization Theorem in One Dimension.- II.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation.- II.10 Exchange of Stability at a Double Point.- II.1 1 Exchange of Stability at a Double Point for Problems Reduced to Local Form.- II.12 Exchange of Stability at a Cusp Point.- II.13 Exchange of Stability at a Triple Point.- II.14 Global Properties of Stability of Isolated Solutions.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- III.1 The Structure of Problems Which Break Double-Point Bifurcation.- III.2 The Implicit Function Theorem and the Saddle Surface Breaking Bifurcation.- III.3 Examples of Isolated Solutions Which Break Bifurcation.- III.4 Iterative Procedures for Finding Solutions.- III.5 Stability of Solutions Which Break Bifurcation.- III.6 Isolas.- Exercise.- Notes.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- IV.1 Eigenvalues and Eigenvectors of an n x n Matrix.- IV.2 Algebraic and Geometric Multiplicity—The Riesz Index.- IV.3 The Adjoint Eigenvalue Problem.- IV.4 Eigenvalues and Eigenvectors of a 2 x 2 Matrix.- 4.1 Eigenvalues.- 4.2 Eigenvectors.- 4.3 Algebraically Simple Eigenvalues.- 4.4 Algebraically Double Eigenvalues.- 4.4.1 Riesz Index 1.- 4.4.2 Riesz Index 2.- IV.5 The Spectral Problem and Stability of the Solution u = 0 in ?n.- IV.6 Nodes, Saddles, and Foci.- IV.7 Criticality and Strict Loss of Stability.- Appendix IV.I Biorthogonality for Generalized Eigenvectors.- Appendix IV.2 Projections.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- V.1 The Form of Steady Bifurcating Solutions and Their Stability.- V.2 Necessary Conditions for the Bifurcation of Steady Solutions.- V.3 Bifurcation at a Simple Eigenvalue.- V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue.- V.5 Bifurcation at a Double Eigenvalue of Index Two.- V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue of Index Two.- V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a Double Eigenvalue of Index One (Semi-Simple).- V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple Double Eigenvalue.- V.9 Examples of Stability Analysis at a Double Semi-Simple (Index-One) Eigenvalue.- V.10 Saddle-Node Bifurcation.- Appendix V.1 Implicit Function Theorem for a System of Two Equations in Two Unknown Functions of One Variable.- Exercises.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VI.1 The Evolution Equation and the Spectral Problem.- VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude.- VI.3 ?1 and ?1 in Projection.- VI.4 Stability of the Bifurcating Solution.- VI.5 The Extra Little Part for ?1 in Projection.- V1.6 Projections of Higher-Dimensional Problems.- VI.7 The Spectral Problem for the Stability of u = 0.- VI.8 The Spectral Problem and the Laplace Transform.- VI.9 Projections into ?1.- VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory).- VI.1 1 The Method of Projection at a Double Eigenvalue of Index Two.- VI.12 The Method of Projection at a Double Semi-Simple Eigenvalue.- VI.13 Examples of the Method of Projection.- VI.14 Symmetry and Pitchfork Bifurcation.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VII.1 The Structure of the Two-Dimensional Problem Governing Hopf Bifurcation.- VII.2 Amplitude Equation for Hopf Bifurcation.- VII.3 Series Solution.- VII.4 Equations Governing the Taylor Coefficients.- VII.5 Solvability Conditions (the Fredholm Alternative).- VII.6 Floquet Theory.- 6.1 Floquet Theory in ?1.- 6.2 Floquet Theory in ?2 and ?n.- VII.7 Equations Governing the Stability of the Periodic Solutions.- VII.8 The Factorization Theorem.- VII.9 Interpretation of the Stability Result.- Example.- VIII Bifurcation of Periodic Solutions in the General Case.- VIII.1 Eigenprojections of the Spectral Problem.- VIII.2 Equations Governing the Projection and the Complementary Projection.- VIII.3 The Series Solution Using the Fredholm Alternative.- VIII.4 Stability of the Hopf Bifurcation in the General Case.- VIII.5 Systems with Rotational Symmetry.- Examples.- Notes.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- Notation.- IX.1 Definition of the Problem of Subharmonic Bifurcation.- IX.2 Spectral Problems and the Eigenvalues ?( µ).- IX.3 Biorthogonality.- IX.4 Criticality.- IX.S The Fredholm Alternative for J( µ) —?( µ)and a Formula Expressing the Strict Crossing (IX.20).- IX.6 Spectral Assumptions.- IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality.- IX.8 The Operator $$\mathbb{J}$$ and its Eigenvectors.- IX.9 The Adjoint Operator $${{\mathbb{J}}^{*}}$$ Biorthogonality, Strict Crossing, and the Fredholm Alternative for $$\mathbb{J}$$.- IX.10 The Amplitude ?and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions.- IX.11 The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to ?at ? =0.- IX.12 Bifurcation and Stability of T-Periodic and 2 T-Periodic Solutions.- IX.13 Bifurcation and Stability of n T-Periodic Solutions with n >2.- IX.14 Bifurcation and Stability of 3T-Periodic Solutions.- IX.15 Bifurcation of 4 T-Periodic Solutions.- IX.16 Stability of 4 T-Periodic Solutions.- IX.17 Nonexistence of Higher-Order Subharmonic Solutions and Weak Resonance.- IX.18 Summary of Results About Subharmonic Bifurcation.- IX.19 Imperfection Theory with a Periodic Imperfection.- Exercises.- IX.20 Saddle-Node Bifurcation of T-Periodic Solutions.- IX.21 General Remarks About Subharmonic Bifurcations.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- X.1 Decomposition of the Solution and Amplitude Equation.- Exercise.- X.2 Derivation of the Amplitude Equation.- X.3 The Normal Equations in Polar Coordinates.- X.4 The Torus and Trajectories on the Torus in the Irrational Case.- X.5 The Torus and Trajectories on the Torus When ?0T/2? Is a Rational Point of Higher Order (n?5).- X.6 The Form of the Torus in the Case n =5.- X.7 Trajectories on the Torus When n =5.- X.8 The Form of the Torus When n >5.- X.9 Trajectories on the Torus When n?5.- X.10 Asymptotically Quasi-Periodic Solutions.- X.11 Stability of the Bifurcated Torus.- X.12 Subharmonic Solutions on the Torus.- X.13 Stability of Subharmonic Solutions on the Torus.- X.14 Frequency Locking.- Appendix X.1 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Irrational Points Using the Method of Two Times, Power Series, and the Fredholm Alternative.- Appendix X.2 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times.- Exercise.- Notes.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case.- Notation.- XI.1 Spectral Problems.- XI.2 Criticality and Rational Points.- XI.3 Spectral Assumptions About J0.- XI.4 Spectral Assumptions About $$\mathbb{J}$$ in the Rational Case.- XI.5 Strict Loss of Stability at a Simple Eigenvalue of J0.- XI.6 Strict Loss of Stability at a Double Semi-Simple Eigenvalue of J0.- XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two.- XI.8 Formulation of the Problem of Subharmonic Bifurcation of Periodic Solutions of Autonomous Problems.- XI.9 The Amplitude of the Bifurcating Solution.- XI.10 Power-Series Solutions of the Bifurcation Problem.- XI.11 Subharmonic Bifurcation When n =2.- XI.12 Subharmonic Bifurcation When n >2.- XI.13 Subharmonic Bifurcation When n = 1in the Semi-Simple Case.- XI.14 “Subharmonic” Bifurcation When n =1 in the Case When Zero is an Index-Two Double Eigenvalue of Jo.- XI.15 Stability of Subharmonic Solutions.- XI.16 Summary of Results About Subharmonic Bifurcation in the Autonomous Case.- XI.17 Amplitude Equations.- XI.18 Amplitude Equations for the Cases n?3 or ?0/?0Irrational.- XI.19 Bifurcating Tori. Asymptotically Quasi-Periodic Solutions.- XI.20 Period Doubling n =2.- XI.21 Pitchfork Bifurcation of Periodic Orbits in the Presence of Symmetry n = 1.- Exercises.- XI.22 Rotationally Symmetric Problems.- Exercise.- XII Stability and Bifurcation in Conservative Systems.- XII.1 The Rolling Ball.- XII.2 Euler Buckling.- Exercises.- XII.3 Some Remarks About Spectral Problems for Conservative Systems.- XII.4 Stability and Bifurcation of Rigid Rotation of Two Immiscible Liquids.- Steady Rigid Rotation of Two Fluids.

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Book SynopsisEquations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations.Trade ReviewJ.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations "This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples."—MATHEMATIKATable of ContentsI: Dimension One.- 1. Scalar Autonomous Equations.- 1.1. Existence and Uniqueness.- 1.2. Geometry of Flows.- 1.3. Stability of Equilibria.- 1.4. Equations on a Circle.- 2. Elementary Bifurcations.- 2.1. Dependence on Parameters - Examples.- 2.2. The Implicit Function Theorem.- 2.3. Local Perturbations Near Equilibria.- 2.4. An Example on a Circle.- 2.5. Computing Bifurcation Diagrams.- 2.6. Equivalence of Flows.- 3. Scalar Maps.- 3.1. Euler’s Algorithm and Maps.- 3.2. Geometry of Scalar Maps.- 3.3. Bifurcations of Monotone Maps.- 3.4. Period-doubling Bifurcation.- 3.5. An Example: The Logistic Map.- II: Dimension One and One Half.- 4. Scalar Nonautonomous Equations.- 4.1. General Properties of Solutions.- 4.2. Geometry of Periodic Equations.- 4.3. Periodic Equations on a Cylinder.- 4.4. Examples of Periodic Equations.- 4.5. Stability of Periodic Solutions.- 5. Bifurcation of Periodic Equations.- 5.1. Bifurcations of Poincaré Maps.- 5.2. Stability of Nonhyperbolic Periodic Solutions.- 5.3. Perturbations of Vector Fields.- 6. On Tori and Circles.- 6.1. Differential Equations on a Torus.- 6.2. Rotation Number.- 6.3. An Example: The Standard Circle Map.- III: Dimension Two.- 7. Planar Autonomous Systems.- 7.1. “Natural” Examples of Planar Systems.- 7.2. General Properties and Geometry.- 7.3. Product Systems.- 7.4. First Integrals and Conservative Systems.- 7.5. Examples of Elementary Bifurcations.- 8. Linear Systems.- 8.1. Properties of Solutions of Linear Systems.- 8.2. Reduction to Canonical Forms.- 8.3. Qualitative Equivalence in Linear Systems.- 8.4. Bifurcations in Linear Systems.- 8.5. Nonhomogeneous Linear Systems.- 8.6. Linear Systems with 1-periodic Coefficients.- 9. Near Equilibria.- 9.1. Asymptotic Stability from Linearization.- 9.2. Instability from Linearization.- 9.3. Liapunov Functions.- 9.4. An Invariance Principle.- 9.5. Preservation of a Saddle.- 9.6. Flow Equivalence Near Hyperbolic Equilibria.- 9.7. Saddle Connections.- 10. In the Presence of a Zero Eigenvalue.- 10.1. Stability.- 10.2. Bifurcations.- 10.3. Center Manifolds.- 11. In the Presence of Purely Imaginary Eigenvalues.- 11.1. Stability.- 11.2. Poincaré-Andronov-Hopf Bifurcation.- 11.3. Computing Bifurcation Curves.- 12. Periodic Orbits.- 12.1. Poincaré-Bendixson Theorem.- 12.2. Stability of Periodic Orbits.- 12.3. Local Bifurcations of Periodic Orbits.- 12.4. A Homoclinic Bifurcation.- 13. All Planar Things Considered.- 13.1. Structurally Stable Vector Fields.- 13.2. Dissipative Systems.- 13.3. One-parameter Generic Bifurcations.- 13.4. Bifurcations in the Presence of Symmetry.- 13.5. Local Two-parameter Bifurcations.- 14- Conservative and Gradient Systems.- 14.1. Second-order Conservative Systems.- 14.2. Bifurcations in Conservative Systems.- 14.3. Gradient Vector Fields.- 15. Planar Maps.- 15.1. Linear Maps.- 15.2. Near Fixed Points.- 15.3. Numerical Algorithms and Maps.- 15.4. Saddle Node and Period Doubling.- 15.5. Poincaré-Andronov-Hopf Bifurcation.- 15.6. Area-preserving Maps.- IV: Higher Dimensions.- 16. Dimension Two and One Half.- 16.1. Forced Van der Pol.- 16.2. Forced Duffing.- 16.3. Near a Transversal Homoclinic Point.- 16.4. Forced and Damped Duffing.- 17. Dimension Three.- 17.1. Period Doubling.- 17.2. Bifurcation to Invariant Torus.- 17.3. Silnikov Orbits.- 17.4. The Lorenz Equations.- 18. Dimension Four.- 18.1. Integrable Hamiltonians.- 18.2. A Nonintegrable Hamiltonian.- Farewell.- APPENDIX: A Catalogue of Fundamental Theorems.- References.

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Book SynopsisThis is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory.Table of Contents1 Geometry in Regions of a Space. Basic Concepts.- §1. Co-ordinate systems.- 1.1. Cartesian co-ordinates in a space.- 1.2. Co-ordinate changes.- §2. Euclidean space.- 2.1. Curves in Euclidean space.- 2.2. Quadratic forms and vectors.- §3. Riemannian and pseudo-Riemannian spaces.- 3.1. Riemannian metrics.- 3.2. The Minkowski metric.- §4. The simplest groups of transformations of Euclidean space.- 4.1. Groups of transformations of a region.- 4.2. Transformations of the plane.- 4.3. The isometries of 3-dimensional Euclidean space.- 4.4. Further examples of transformation groups.- 4.5. Exercises.- §5. The Serret—Frenet formulae.- 5.1. Curvature of curves in the Euclidean plane.- 5.2. Curves in Euclidean 3-space. Curvature and torsion.- 5.3. Orthogonal transformations depending on a parameter.- 5.4. Exercises.- §6. Pseudo-Euclidean spaces.- 6.1. The simplest concepts of the special theory of relativity.- 6.2. Lorentz transformations.- 6.3. Exercises.- 2 The Theory of Surfaces.- §7. Geometry on a surface in space.- 7.1. Co-ordinates on a surface.- 7.2. Tangent planes.- 7.3. The metric on a surface in Euclidean space.- 7.4. Surface area.- 7.5. Exercises.- §8. The second fundamental form.- 8.1. Curvature of curves on a surface in Euclidean space.- 8.2. Invariants of a pair of quadratic forms.- 8.3. Properties of the second fundamental form.- 8.4. Exercises.- §9. The metric on the sphere.- §10. Space-like surfaces in pseudo-Euclidean space.- 10.1. The pseudo-sphere.- 10.2. Curvature of space-like curves in $$ \mathbb{R}_1^3 $$.- §11. The language of complex numbers in geometry.- 11.1. Complex and real co-ordinates.- 11.2. The Hermitian scalar product.- 11.3. Examples of complex transformation groups.- §12. Analytic functions.- 12.1. Complex notation for the element of length, and for the differential of a function.- 12.2. Complex co-ordinate changes.- 12.3. Surfaces in complex space.- §13. The conformal form of the metric on a surface.- 13.1. Isothermal co-ordinates. Gaussian curvature in terms of conformal co-ordinates.- 13.2. Conformal form of the metrics on the sphere and the Lobachevskian plane.- 13.3. Surfaces of constant curvature.- 13.4. Exercises.- §14. Transformation groups as surfaces in N-dimensional space.- 14.1. Co-ordinates in a neighbourhood of the identity.- 14.2. The exponential function with matrix argument.- 14.3. The quaternions.- 14.4. Exercises.- §15. Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions.- 3 Tensors: The Algebraic Theory.- §16. Examples of tensors.- §17. The general definition of a tensor.- 17.1. The transformation rule for the components of a tensor of arbitrary rank.- 17.2. Algebraic operations on tensors.- 17.3. Exercises.- §18. Tensors of type (0, k).- 18.1. Differential notation for tensors with lower indices only.- 18.2. Skew-symmetric tensors of type (0, k).- 18.3. The exterior product of differential forms. The exterior algebra.- 18.4. Skew-symmetric tensors of type (k, 0) (polyvectors). Integrals with respect to anti-commuting variables.- 18.5. Exercises.- §19. Tensors in Riemannian and pseudo-Riemannian spaces.- 19.1. Raising and lowering indices.- 19.2. The eigenvalues of a quadratic form.- 19.3. The operator ?.- 19.4. Tensors in Euclidean space.- 19.5. Exercises.- §20. The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space. Examples of invariant tensors.- §21. Rank 2 tensors in pseudo-Euclidean space, and their eigenvalues.- 21.1. Skew-symmetric tensors. The invariants of an electromagnetic field.- 21.2. Symmetric tensors and their eigenvalues. The energy-momentum tensor of an electromagnetic field.- §22. The behaviour of tensors under mappings.- 22.1. The general operation of restriction of tensors with lower indices.- 22.2. Mappings of tangent spaces.- §23. Vector fields.- 23.1. One-parameter groups of diffeomorphisms.- 23.2. The exponential function of a vector field.- 23.3. The Lie derivative.- 23.4. Exercises.- §24. Lie algebras.- 24.1. Lie algebras and vector fields.- 24.2. The fundamental matrix Lie algebras.- 24.3. Linear vector fields.- 24.4. Left-invariant fields defined on transformation groups.- 24.5. Invariant metrics on a transformation group.- 24.6. The classification of the 3-dimensional Lie algebras.- 24.7. The Lie algebras of the conformal groups.- 24.8. Exercises.- 4 The Differential Calculus of Tensors.- §25. The differential calculus of skew-symmetric tensors.- 25.1. The gradient of a skew-symmetric tensor.- 25.2. The exterior derivative of a form.- 25.3. Exercises.- §26. Skew-symmetric tensors and the theory of integration.- 26.1. Integration of differential forms.- 26.2. Examples of integrals of differential forms.- 26.3. The general Stokes formula. Examples.- 26.4. Proof of the general Stokes formula for the cube.- 26.5. Exercises.- §27. Differential forms on complex spaces.- 27.1. The operators d? and d?.- 27.2. Kählerian metrics. The curvature form.- §28. Covariant differentiation.- 28.1. Euclidean connexions.- 28.2. Covariant differentiation of tensors of arbitrary rank.- §29. Covariant differentiation and the metric.- 29.1. Parallel transport of vector fields.- 29.2. Geodesics.- 29.3. Connexions compatible with the metric.- 29.4. Connexions compatible with a complex structure (Hermitian metric).- 29.5. Exercises.- §30. The curvature tensor.- 30.1. The general curvature tensor.- 30.2. The symmetries of the curvature tensor. The curvature tensor defined by the metric.- 30.3. Examples: The curvature tensor in spaces of dimensions 2 and 3; the curvature tensor of transformation groups.- 30.4. The Peterson—Codazzi equations. Surfaces of constant negative curvature, and the “sine—Gordon” equation.- 30.5. Exercises.- 5 The Elements of the Calculus of Variations.- §31. One-dimensional variational problems.- 31.1. The Euler—Lagrange equations.- 31.2. Basic examples of functional.- §32. Conservation laws.- 32.1. Groups of transformations preserving a given variational problem.- 32.2. Examples. Applications of the conservation laws.- §33. Hamiltonian formalism.- 33.1. Legendre’s transformation.- 33.2. Moving co-ordinate frames.- 33.3. The principles of Maupertuis and Fermat.- 33.4. Exercises.- §34. The geometrical theory of phase space.- 34.1. Gradient systems.- 34.2. The Poisson bracket.- 34.3. Canonical transformations.- 34.4. Exercises.- §35. Lagrange surfaces.- 35.1. Bundles of trajectories and the Hamilton—Jacobi equation.- 35.2. Hamiltonians which are first-order homogeneous with respect to the momentum.- §36. The second variation for the equation of the geodesics.- 36.1. The formula for the second variation.- 36.2. Conjugate points and the minimality condition.- 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants.- §37. The simplest higher-dimensional variational problems.- 37.1. The Euler—Lagrange equations.- 37.2. The energy-momentum tensor.- 37.3. The equations of an electromagnetic field.- 37.4. The equations of a gravitational field.- 37.5. Soap films.- 37.6. Equilibrium equation for a thin plate.- 37.7. Exercises.- §38. Examples of Lagrangians.- §39. The simplest concepts of the general theory of relativity.- §40. The spinor representations of the groups SO(3) and O(3, 1). Dirac’s equation and its properties.- 40.1. Automorphisms of matrix algebras.- 40.2. The spinor representation of the group SO(3).- 40.3. The spinor representation of the Lorentz group.- 40.4. Dirac’s equation.- 40.5. Dirac’s equation in an electromagnetic field. The operation of charge conjugation.- §41. Covariant differentiation of fields with arbitrary symmetry.- 41.1. Gauge transformations. Gauge-invariant Lagrangians.- 41.2. The curvature form.- 41.3. Basic examples.- §42. Examples of gauge-invariant functionals. Maxwell’s equations and the Yang—Mills equation. Functionals with identically zero variational derivative (characteristic classes).

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