Fritz John - Böcker

From the reviews of the earlier editions: “An excellent second-reading text.  Should be accessible to any mathematician.  Highly recommended.” – The Mathematical Gazette“An excellent text upon which to build and introduction to partial differential equations at the graduate level.  In addition…recommended for the nonspecialist.” – American Mathematical MonthlyPartial Differential Equations is a very well accepted introduction to the subject.  In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis.  Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation with solutions.

The description for this book, Contributions to the Theory of Partial Differential Equations. (AM-33), will be forthcoming.

The mathematical works of Fritz John whose deep and original ideas have had a great influence on the development of various fields in mathema­ tical analysis are made available with these volumes. His works are certainly well known to the experts, but knowledge of his contributions may not have spread as widely as it should have. For example, the concept of functions of bounded mean oscillations plays a central role in harmonic analysis today, but it is perhaps less known that this class of functions was introduced by John as early as 1961, motivated by his work in elasticity theory. With the publication of this collection, a wider circle of mathematicians will become familiar with, and appreciate, the fertile ideas of Fritz John. The organization of these two volumes was undertaken in consultation with the author. It was decided not to present the papers in chronological order, but rather to subdivide them into ten sections representing different mathematical topics to which John has contributed. Commentaries made by experts in the fields are appended to each section. Since the division into sec­ tions could, of course, not be made sharply, there are several overlaps. For instance, the comments of Louis Nirenberg refer to Elasticity Theory VI, Geometric Inequalities VIII, and Functions of Bounded Mean Oscillations IX. To help the reader, cross-references and remarks by the author will be found at the end of each section.

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.

From the reviews of the earlier editions: “An excellent second-reading text.  Should be accessible to any mathematician.  Highly recommended.” – The Mathematical Gazette“An excellent text upon which to build and introduction to partial differential equations at the graduate level.  In addition…recommended for the nonspecialist.” – American Mathematical MonthlyPartial Differential Equations is a very well accepted introduction to the subject.  In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis.  Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation with solutions.

From the reviews: "Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)" The Mathematical Gazette (75.1991.471)

From the reviews: "These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that."Newsletter on Computational and Applied Mathematics, 1991"...one of the best textbooks introducing several generations of mathematicians to higher mathematics...This excellent book is highly recommended both to instructors and students." Acta Scientiarum Mathematicarum, 1991