Geometry Revealed

A Jacob's Ladder to Modern Higher Geometry

Author: Marcel Berger

Publisher: Springer Science & Business Media

ISBN: 9783540709978

Category: Mathematics

Page: 831

View: 2167

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.
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Geometry Revealed

A Jacob's Ladder to Modern Higher Geometry

Author: Marcel Berger

Publisher: Springer

ISBN: 9783662501221

Category:

Page: 848

View: 8588

This inviting and visually rich book is the latest work of well known mathematician Marcel Berger. He aims to demonstrate to readers the renewed spirit of geometry, and reveals that even so-called "elementary" geometry is very much alive.
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Problems in Geometry

Author: Marcel Berger,P. Pansu,J.-P. Berry,X. Saint-Raymond

Publisher: Springer Science & Business Media

ISBN: 1475718365

Category: Mathematics

Page: 268

View: 3978

The textbook Geometry, published in French by CEDICjFernand Nathan and in English by Springer-Verlag (scheduled for 1985) was very favorably re ceived. Nevertheless, many readers found the text too concise and the exercises at the end of each chapter too difficult, and regretted the absence of any hints for the solution of the exercises. This book is intended to respond, at least in part, to these needs. The length of the textbook (which will be referred to as [B] throughout this book) and the volume of the material covered in it preclude any thought of publishing an expanded version, but we considered that it might prove both profitable and amusing to some of our readers to have detailed solutions to some of the exercises in the textbook. At the same time, we planned this book to be independent, at least to a certain extent, from the textbook; thus, we have provided summaries of each of its twenty chapters, condensing in a few pages and under the same titles the most important notions and results,used in the solution of the problems. The statement of the selected problems follows each summary, and they are numbered in order, with a reference to the corresponding place in [B]. These references are not meant as indications for the solutions of the problems. In the body of each summary there are frequent references to [B], and these can be helpful in elaborating a point which is discussed too cursorily in this book.
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A Panoramic View of Riemannian Geometry

Author: Marcel Berger

Publisher: Springer Science & Business Media

ISBN: 3642182453

Category: Mathematics

Page: 824

View: 1614

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS
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Elements of Algebra

Geometry, Numbers, Equations

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 1475739761

Category: Mathematics

Page: 184

View: 5669

Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.
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Advanced Calculus

A Geometric View

Author: James J. Callahan

Publisher: Springer Science & Business Media

ISBN: 9781441973320

Category: Mathematics

Page: 526

View: 991

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. The book incorporates modern computational tools to give visualization real power. Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books. This is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.
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Geometry: Euclid and Beyond

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 0387226761

Category: Mathematics

Page: 528

View: 7518

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.
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Geometry I

Author: Marcel Berger,M. Cole

Publisher: Springer Science & Business Media

ISBN: 9783540116585

Category: Mathematics

Page: 432

View: 3012

The first part of a two-volume text providing a readable and lively presentation of large parts of geometry in the classical sense, this book appeals systematically to the reader's intuition and vision, and illustrates the mathematical text with many figures.
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Elegant Chaos

Author: N.A

Publisher: N.A

ISBN: 9814468673

Category:

Page: N.A

View: 7084

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Geometry

Seeing, Doing, Understanding

Author: Harold R. Jacobs

Publisher: Macmillan

ISBN: 9780716743613

Category: Mathematics

Page: 780

View: 6451

Harold Jacobs's Geometry created a revolution in the approach to teaching this subject, one that gave rise to many ideas now seen in the NCTM Standards. Since its publication nearly one million students have used this legendary text. Suitable for either classroom use or self-paced study, it uses innovative discussions, cartoons, anecdotes, examples, and exercises that unfailingly capture and hold student interest. This edition is the Jacobs for a new generation. It has all the features that have kept the text in class by itself for nearly 3 decades, all in a thoroughly revised, full-color presentation that shows today's students how fun geometry can be. The text remains proof-based although the presentation is in the less formal paragraph format. The approach focuses on guided discovery to help students develop geometric intuition.
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Man and His Symbols

Author: C. G. Jung

Publisher: Dell

ISBN: 0307800555

Category: Social Science

Page: 320

View: 7271

Man and His Symbols owes its existence to one of Jung's own dreams. The great psychologist dreamed that his work was understood by a wide public, rather than just by psychiatrists, and therefore he agreed to write and edit this fascinating book. Here, Jung examines the full world of the unconscious, whose language he believed to be the symbols constantly revealed in dreams. Convinced that dreams offer practical advice, sent from the unconscious to the conscious self, Jung felt that self-understanding would lead to a full and productive life. Thus, the reader will gain new insights into himself from this thoughtful volume, which also illustrates symbols throughout history. Completed just before his death by Jung and his associates, it is clearly addressed to the general reader. Praise for Man and His Symbols “This book, which was the last piece of work undertaken by Jung before his death in 1961, provides a unique opportunity to assess his contribution to the life and thought of our time, for it was also his firsat attempt to present his life-work in psychology to a non-technical public. . . . What emerges with great clarity from the book is that Jung has done immense service both to psychology as a science and to our general understanding of man in society, by insisting that imaginative life must be taken seriously in its own right, as the most distinctive characteristic of human beings.”—Guardian “Straighforward to read and rich in suggestion.”—John Barkham, Saturday Review Syndicate “This book will be a resounding success for those who read it.”—Galveston News-Tribune “A magnificent achievement.”—Main Currents “Factual and revealing.”—Atlanta Times
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Where Mathematics Comes from

How the Embodied Mind Brings Mathematics Into Being

Author: George Lakoff,Rafael E. Núñez

Publisher: Basic Books (AZ)

ISBN: N.A

Category: Mathematics

Page: 493

View: 1702

Provides an in-depth analysis of the cognitive science of mathematical ideas that argues that conceptual metaphor plays a definitive role in mathematical ideas, exploring such concepts as arithmetic, algebra, sets, logic, and infinity. 20,000 first printing.
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Concrete Mathematics

A Foundation for Computer Science

Author: Ronald L. Graham,Donald Ervin Knuth,Oren Patashnik

Publisher: Addison-Wesley Professional

ISBN: 9780201558029

Category: Computers

Page: 657

View: 4344

This book, updated and improved, introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. It is an indispensable text and reference, not only for computer scientists but for all technical professionals in virtually every discipline.
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Gnosis of the Cosmic Christ

A Gnostic Christian Kabbalah

Author: Tau Malachi

Publisher: Llewellyn Worldwide Limited

ISBN: 9780738705910

Category: Body, Mind & Spirit

Page: 400

View: 4624

First presents the basic ideas of the Christian Kabbalah--which is about experiencing God and moving toward God--then goes to a more in-depth exploration of the Tree of Life. Original.
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Observations and Predictions of Eclipse Times by Early Astronomers

Author: J.M. Steele

Publisher: Springer Science & Business Media

ISBN: 9401595283

Category: History

Page: 324

View: 4896

Eclipses have long been seen as important celestial phenomena, whether as omens affecting the future of kingdoms, or as useful astronomical events to help in deriving essential parameters for theories of the motion of the moon and sun. This is the first book to collect together all presently known records of timed eclipse observations and predictions from antiquity to the time of the invention of the telescope. In addition to cataloguing and assessing the accuracy of the various records, which come from regions as diverse as Ancient Mesopotamia, China, and Europe, the sources in which they are found are described in detail. Related questions such as what type of clocks were used to time the observations, how the eclipse predictions were made, and how these prediction schemes were derived from the available observations are also considered. The results of this investigation have important consequences for how we understand the relationship between observation and theory in early science and the role of astronomy in early cultures, and will be of interest to historians of science, astronomers, and ancient and medieval historians.
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Geometry II

Author: Marcel Berger

Publisher: Springer Science & Business Media

ISBN: 9783540170150

Category: Mathematics

Page: 406

View: 1725

This is the second of a two-volume textbook that provides a very readable and lively presentation of large parts of geometry in the classical sense. For each topic the author presents a theorem that is esthetically pleasing and easily stated, although the proof may be quite hard and concealed. Yet another strong trait of the book is that it provides a comprehensive and unified reference source for the field of geometry in the full breadth of its subfields and ramifications.
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Additive Combinatorics

Author: Terence Tao,Van H. Vu

Publisher: Cambridge University Press

ISBN: 1139458345

Category: Mathematics

Page: N.A

View: 6364

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
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Beautiful Geometry

Author: Eli Maor,Eugen Jost

Publisher: Princeton University Press

ISBN: 1400848334

Category: Mathematics

Page: 208

View: 3028

If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important branches of mathematics.
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Geometry of Curves and Surfaces with MAPLE

Author: Vladimir Rovenski

Publisher: Springer Science & Business Media

ISBN: 1461221285

Category: Mathematics

Page: 310

View: 9370

This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization of curves and surfaces. The author systematically examines such powerful tools as 2-D and 3-D animation of geometric images, transformations, shadows, and colors, and then further studies more complex problems in differential geometry. Well-illustrated with more than 350 figures---reproducible using Maple programs in the book---the work is devoted to three main areas: curves, surfaces, and polyhedra. Pedagogical benefits can be found in the large number of Maple programs, some of which are analogous to C++ programs, including those for splines and fractals. To avoid tedious typing, readers will be able to download many of the programs from the Birkhauser web site. Aimed at a broad audience of students, instructors of mathematics, computer scientists, and engineers who have knowledge of analytical geometry, i.e., method of coordinates, this text will be an excellent classroom resource or self-study reference. With over 100 stimulating exercises, problems and solutions, {\it Geometry of Curves and Surfaces with Maple} will integrate traditional differential and non- Euclidean geometries with more current computer algebra systems in a practical and user-friendly format.
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The Interpretation of Cultures

Author: Clifford Geertz

Publisher: Basic Books

ISBN: 0465093566

Category: Social Science

Page: 576

View: 5652

In The Interpretation of Cultures, the most original anthropologist of his generation moved far beyond the traditional confines of his discipline to develop an important new concept of culture. This groundbreaking book, winner of the 1974 Sorokin Award of the American Sociological Association, helped define for an entire generation of anthropologists what their field is ultimately about.
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