For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.

What Is Mathematics? An Elementary Approach to Ideas and Methods

Science & Math

Richard Courant

Herbert Robbins

Mathematics

Oxford University Press

It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this category. It is written in a style which, as the author hopes, expresses adequately the balance and tension between the individuality of mathematical objects and the generality of mathematical methods. The author has been interested in Dirichlet's Principle and its various applications since his days as a student under David Hilbert. Plans for writing a book on these topics were revived when Jesse Douglas' work suggested to him a close connection between Dirichlet's Principle and basic problems concerning minimal sur- faces. But war work and other duties intervened; even now, after much delay, the book appears in a much less polished and complete form than the author would have liked.

Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces

Springer

The new Chapter 1 contains all the fundamental properties of linear differential forms and their integrals. These prepare the reader for the introduction to higher-order exterior differential forms added to Chapter 3. Also found now in Chapter 3 are a new proof of the implicit function theorem by successive approximations and a discus sion of numbers of critical points and of indices of vector fields in two dimensions. Extensive additions were made to the fundamental properties of multiple integrals in Chapters 4 and 5. Here one is faced with a familiar difficulty: integrals over a manifold M, defined easily enough by subdividing M into convenient pieces, must be shown to be inde pendent of the particular subdivision. This is resolved by the sys tematic use of the family of Jordan measurable sets with its finite intersection property and of partitions of unity. In order to minimize topological complications, only manifolds imbedded smoothly into Euclidean space are considered. The notion of "orientation" of a manifold is studied in the detail needed for the discussion of integrals of exterior differential forms and of their additivity properties. On this basis, proofs are given for the divergence theorem and for Stokes's theorem in n dimensions. To the section on Fourier integrals in Chapter 4 there has been added a discussion of Parseval's identity and of multiple Fourier integrals.

Introduction to Calculus and Analysis, Vol. 2 (Classics in Mathematics)

Methods of Mathematical Physics, Vol. 1

Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.

David HILBERT

Physics

John Wiley

Methods of Mathematical Physics Volume II Two 2 Partial Differential Equations

Methods of Mathematical Physics. Volume 2: Partial Differential Equations (v. 2)

Interscience Publishers

“This is the perfect solid-as-they-come, timeless book on the calculus, and most likely it will never be surpassed in this domain.” –Amazon Review This book is intended for anyone who, having passed through an ordinary course of school mathematics, wishes to apply himself to the study of mathematics or its applications to science and engineering, no matter whether he is a student of a university or technical college, a teacher, or an engineer. Courant leads the way straight to useful knowledge, and aims at making the subject easier to grasp, not only by giving proofs step by step, but also by throwing light on the interconnections and purposes of the whole.

Differential and Integral Calculus, Vol. 1, Second Edition

wiley

Introduction to Calculus and Analysis, Vol. 2

John Wiley&Sons, Inc.

D. Hilbert

This Set Contains: Differential and Integral Calculus, Volume 1 by R. Courant; Differential and Integral Calculus, Volume 2 by R. Courant.

Differential and Integral Calculus (2 Volume Set)

Wiley-Interscience

Mathematical development, the author of this text observes, comes about through specific, easily understood problems that require difficult solutions and demand the use of new methods. Richard Courant employs this instructive approach in a text that balances the individuality of mathematical objects with the generality of mathematical methods. Beginning with a discussion of Dirichlet's principle and the boundary-value problem of potential theory, the text proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Succeeding chapters explore the general problem of Douglas and conformal mapping of multiply connected domains, concluding with a survey of minimal surfaces with free boundaries and unstable minimal surfaces.

Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces (Dover Books on Mathematics)

Dover Publications

See the best price to sell, buy, or rent books by Richard Courant. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Richard Courant

See the best price to sell, buy, or rent books by Richard Courant. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

What Is Mathematics? An Elementary Approach to Ideas and Methods

Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces

Introduction to Calculus and Analysis, Vol. 2 (Classics in Mathematics)

Methods of Mathematical Physics, Vol. 1

Methods of Mathematical Physics. Volume 2: Partial Differential Equations (v. 2)

Differential and Integral Calculus, Vol. 1, Second Edition

Introduction to Calculus and Analysis, Vol. 2

Methods of Mathematical Physics, Vol. 1

Differential and Integral Calculus (2 Volume Set)

Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces (Dover Books on Mathematics)