The latest volume in the New York Times–bestselling physics series explains Einstein’s masterpiece: the general theory of relativity He 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.   Read more

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A fewnew topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. Read more

This volume has consistently been a leading text in its field since its initial release. It has remained in continuous print, reflecting its enduring relevance and significance in scholarly and professional discourse.   One of the most widely used texts in its field, this volume has been continuously in print since its initial 1976 publication. Many examples and exercises, some with hints and answers, enhance the clear, well-written exposition. Prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables.   Comprehensive Coverage: The volume introduces a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. It provides a thorough understanding of key concepts and techniques. Clear Exposition: The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Examples and Exercises: The book includes numerous examples and exercises that enhance the learning experience. These practical applications solidify understanding and provide opportunities for practice and self-assessment. Hints and Answers: Hints and answers to selected problems are provided, allowing readers to check their work and gain further insights into the subject matter. Updated Edition: This second edition has been corrected, revised, and updated by the author, ensuring that the content remains current and relevant. Suitable for Advanced Study: While suitable for advanced undergraduates, this text is also appropriate for graduate students of mathematics. Prerequisites include an undergraduate course in linear algebra and some familiarity with calculus of several variables. About the author: Manfredo P. do Carmo was a Brazilian mathematician and authority in the very active field of differential geometry. He was an emeritus researcher at Rio's National Institute for Pure and Applied Mathematics and the author of Differential Forms and Applications. Read more

Born of the desire to understand the workings of motions of the heavenly bodies, trigonometry gave the ancient Greeks the ability to predict their futures. Most of what we see of the subject in school comes from these heavenly origins; 15th century astronomer Regiomontanus called it "the foot of the ladder to the stars." In this Very Short Introduction Glen Van Brummelen shows how trigonometry connects mathematics to science, and has today become an indispensable tool in predicting cyclic patterns like animal populations and ocean tides. Its historical journey through major cultures such as medieval India and the Islamic World has taken it through disciplines such as geography and even religious practice. Trigonometry has also been a major player in the most startling mathematical developments of the modern world. Its interactions with the concept of infinity led to Taylor and Fourier series, some of the most practical tools of modern science. The birth of complex numbers led to a shocking union of exponential and trigonometric functions, creating the most beautiful formulas and powerful modelling tools in science. Finally, as Van Brummelen shows, trigonometry allows us to explore the strange new worlds of non-Euclidean geometries, opening up bizarre possibilities for the shape of space itself. And indeed, one of those new geometries - spherical - takes us full circle back to ancient Greek astronomers and European navigators, who first used it to chart their ways across the heavens and the earth.ABOUT THE SERIES:The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. Read more

Graph Paper Composition Notebook - 5x5 Grid - 8.5 x 11 - 100 Pages1/5 Inch Graph Ruled Paper: small grid, 5 squares per inch, ideal for math, science and engineering students, professionals, artists and bullet journalists.100 Pages / 50 Sheets of premium white paper with light grey grid lines, offering plenty of space for all your notes, drawings, and ideas.8.5 x 11 Inches (21.59 x 27.94 cm): large standard size notebook.Minimalist, Aesthetic Dusty Rose Pink Cover: Stylish matte finish and durable design that withstands everyday use.Multipurpose UseStudy - High School, College and Grad SchoolProfessional Work - for Scientists, Engineers, Architects, Designers, Researchers, Analysts and LabDetailed and precise graphing, geometry, plotting, problem-solving, sketching, and note-takingArts and CraftsJournalingDrawing and DoodlingDiary entries and list makingCapturing ideas Read more

The third volume in the bestselling physics series cracks open Einstein's special relativity and field theory Physicist Leonard Susskind and data engineer Art Friedman are back. This time, they introduce readers to Einstein's special relativity and Maxwell's classical field theory. Using their typical brand of real math, enlightening drawings, and humor, Susskind and Friedman walk us through the complexities of waves, forces, and particles by exploring special relativity and electromagnetism. It's a must-read for both devotees of the series and any armchair physicist who wants to improve their knowledge of physics' deepest truths. Read more

In general terms, the shape of an object, data set, or image can be de­ fined as the total of all information that is invariant under translations, rotations, and isotropic rescalings. Thus two objects can be said to have the same shape if they are similar in the sense of Euclidean geometry. For example, all equilateral triangles have the same shape, and so do all cubes. In applications, bodies rarely have exactly the same shape within measure­ ment error. In such cases the variation in shape can often be the subject of statistical analysis. The last decade has seen a considerable growth in interest in the statis­ tical theory of shape. This has been the result of a synthesis of a number of different areas and a recognition that there is considerable common ground among these areas in their study of shape variation. Despite this synthesis of disciplines, there are several different schools of statistical shape analysis. One of these, the Kendall school of shape analysis, uses a variety of mathe­ matical tools from differential geometry and probability, and is the subject of this book. The book does not assume a particularly strong background by the reader in these subjects, and so a brief introduction is provided to each of these topics. Anyone who is unfamiliar with this material is advised to consult a more complete reference. As the literature on these subjects is vast, the introductory sections can be used as a brief guide to the literature. Read more

This volume covers local as well as global differential geometry of curves and surfaces. Read more

Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. Read more

Polytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H. S. M. Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information that has been accumulated on them since, especially in the last hundred years. The author, professor of Mathematics, University of Toronto, has contributed much valuable work himself on polytopes and is a well-known authority on them. Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler's formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author's lucid explanations also help to make the text readily comprehensible. Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming, and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study. Read more

Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory. The book can serve as a textbook for undergraduate students and for graduate students in geometry. Read more

A Comprehensive Introduction to Differential Geometry, Volume Two, Third Edition" by Michael Spivak, published by Publish or Perish, Inc. in Houston, Texas, 2025, covers advanced topics in differential geometry. The book follows a semi-historical path, focusing on classical differential geometry in Chapters 3 and 4. Although it lacks problem sets, it compensates with a comprehensive bibliography in the final volume. The book delves into intricate details of differential geometry, providing a thorough understanding of the subject through historical context, theoretical foundations, and advanced mathematical formulations. Key topics include curves in the plane and in space, the curvature of surfaces in space, the curvature of higher-dimensional manifolds, the absolute differential calculus (Ricci calculus), the r operator, the repère mobile (moving frame), and connections in principal bundles. The document section is an excerpt from the book, specifically discussing Gauss' investigations of curved surfaces. It provides detailed explanations and proofs of various geometric properties and theorems related to the curvature of surfaces. The section combines historical context, mathematical rigor, and detailed proofs to explore the fundamental aspects of differential geometry, focusing on Gauss' contributions and Riemann's advancements. Overall, the book is a valuable resource for anyone looking to deepen their understanding of differential geometry, offering a comprehensive and authoritative guide to the subject. Read more

A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition by Michael Spivak is an essential resource for anyone interested in the field of differential geometry. Published by Publish or Perish, Inc. in Houston, Texas, this volume is part of a five-volume series that has been meticulously crafted to bridge the gap between classical and modern differential geometry. This third edition includes significant updates and corrections, along with the inclusion of Gauss' paper in Volume 2. The book provides a thorough introduction to the theory of differentiable manifolds, emphasizing both historical context and modern language. It covers a wide range of topics, including immersion and functions, the tangent bundle, integral curves and vector fields, differential equations, integration of differential forms, inner product on vector spaces, critical points and conditions, exact sequence of complexes and cohomology vector spaces, and Euler characteristic and related theorems. Unique features of this book include detailed explanations and propositions related to integral curves, vector fields, and coordinate systems, comprehensive coverage of solving differential equations and integration of differential forms, in-depth discussion on inner product on vector spaces, including Euclidean and Riemannian metrics, exploration of critical points, variational approach, and geodesics, extensive treatment of exact sequences, cohomology vector spaces, and Euler characteristic, and application of Poincaré duality theorem, Thom class, Euler class, vector fields, and indices. With its rigorous approach and clear presentation, A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition is an invaluable addition to the library of mathematicians, physicists, and anyone with a keen interest in differential geometry. Read more

A Comprehensive Introduction to Differential Geometry, Volume Four, Third Edition" by Michael Spivak, published by Publish or Perish, Inc. in 1999, is an advanced mathematical text focusing on differential geometry. This volume, part of a five-volume series, delves into higher dimensions and codimensions, curves in Riemannian manifolds, and the fundamental equations for submanifolds. Key features of this volume include a detailed exploration of constant curvature manifolds, comprehensive treatment of curves in Riemannian manifolds, and in-depth discussion of the fundamental equations for submanifolds. The book extensively uses orthonormal moving frames and connection forms, providing rigorous proofs and theorems such as Beltrami's Theorem and the Ricci Equations. This volume is ideal for advanced students and researchers in mathematics, providing a thorough understanding of differential geometry's higher-dimensional aspects. It includes detailed mathematical formulations and proofs to support the presented concepts, making it an essential resource for those studying or working in the field of differential geometry. Read more

FROM THE AUTHOR:All legitimate copies of VDGF produced by Princeton University Press are crisply printed on high-quality paper.  If you obtain a shoddily printed copy, it's a fake: please return it and purchase a genuine PUP copy.An inviting, intuitive, and visual exploration of differential geometry and formsVisual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide new geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to Differential Forms that treats advanced topics in an intuitive and geometrical manner.Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology.The final act provides an intuitive, geometrical introduction to Differential Forms, elucidating such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book, providing a self-contained introduction to Differential Forms.Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be understood and taught. Read more

A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts. Read more

"A must-read.”―Avi Loeb, New York Times–bestselling author of Extraterrestrial One of the preeminent mathematicians of the past half century shows how physics and math were combined to give us the theory of gravity and the dizzying array of ideas and insights that has come from it  Mathematics is far more than just the language of science. It is a critical underpinning of nature. The famed physicist Albert Einstein demonstrated this in 1915 when he showed that gravity—long considered an attractive force between massive objects—was actually a manifestation of the curvature, or geometry, of space and time. But in making this towering intellectual leap, Einstein needed the help of several mathematicians, including Marcel Grossmann, who introduced him to the geometrical framework upon which his theory rest.   In The Gravity of Math, Steve Nadis and Shing-Tung Yau consider how math can drive and sometimes even anticipate discoveries in physics. Examining phenomena like black holes, gravitational waves, and the Big Bang, Nadis and Yau ask: Why do mathematical statements, derived solely from logic, provide the best descriptions of our physical world?  The Gravity of Math offers an insightful and compelling look into the power of mathematics—whose reach, like that of gravity, can extend to the edge of the universe. Read more

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

The beaver's tooth and the tiger's claw. Sunflowers and seashells. Fractals, Fibonacci sequences, and logarithmic spirals. These diverse forms of nature and mathematics are united by a common factor: all involve self-repeating shapes, or gnomons. Almost two thousand years ago, Hero of Alexandria defined the gnomon as that form which, when added to some form, results in a new form, similar to the original. In a spiral seashell, for example, we see that each new section of growth (the gnomon) resembles its predecessor and maintains the shell's overall shape. Inspired by Hero, Midhat Gazalé--a fellow native of Alexandria--explains the properties of gnomons, traces their long and colorful history in human thought, and explores the mathematical and geometrical marvels they make possible.Gazalé is a man of wide-ranging interests and accomplishments. He is a mathematician and engineer who teaches at the University of Paris and whose business career lifted him to the Presidency of AT&T-France. He has a passion for numbers that is clear on every page, as he combines elegant mathematical explanations with compelling anecdotes and a rich variety of illustrations. He begins by explaining the basic properties of gnomons and tracing the term--which originally meant "that which allows one to know"--to ancient Egyptian and Greek timekeeping. Gazalé examines figurate numbers, which inspired the Greek notions of gnomon and number similarity. He introduces us to continued fractions and guides us through the intricacies of Fibonacci sequences, ladder networks, whorled figures, the famous "golden number," logarithmic spirals, and fractals. Along the way, he draws our attention to a host of intriguing and eccentric concepts, shapes, and numbers, from a complex geometric game invented by the nineteenth-century mathematician William Hamilton to a peculiar triangular shape that Gazalé terms the "winkle." Throughout, the book brims with original observations and research, from the presentation of a cousin of the "golden rectangle" that Gazalé calls the "silver pentagon" to the introduction of various new fractal figures and the coining of the term "gnomonicity" for the concept of self-similarity.This is an erudite, engaging, and beautifully produced work that will appeal to anyone interested in the wonders of geometry and mathematics, as well as to enthusiasts of mathematical puzzles and recreations. Read more

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'. Read more