Absolute analysis (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete)

The first edition of this book, published in German, came into being as the result of lectures which the authors held over a period of several years since 1953 at the Universities of Helsinki and Zurich. The Introduction, which follows, provides information on what moti­ vated our presentation of an absolute, coordinate- and dimension-free infinitesimal calculus. Little previous knowledge is presumed of the reader. It can be recom­ mended to students familiar with the usual structure, based on co­ ordinates, of the elements of analytic geometry, differential and integral calculus and of the theory of differential equations. We are indebted to H. Keller, T. Klemola, T. Nieminen, Ph. Tondeur and K. 1. Virtanen, who read our presentation in our first manuscript, for important critical remarks. The present new English edition deviates at several points from the first edition (d. Introduction). Professor I. S. Louhivaara has from the beginning to the end taken part in the production of the new edition and has advanced our work by suggestions on both content and form. For his important support we wish to express our hearty thanks. We are indebted also to W. Greub and to H. Haahti for various valuable remarks. Our manuscript for this new edition has been translated into English by Doctor P. Emig. We express to him our gratitude for his careful interest and skillful attention during this work. The first edition of this book, published in German, came into being as the result of lectures which the authors held over a period of several years since 1953 at the Universities of Helsinki and Zurich. The Introduction, which follows, provides information on what motiƯ vated our presentation of an absolute, coordinate- and dimension-free infinitesimal calculus. Little previous knowledge is presumed of the reader. It can be recomƯ mended to students familiar with the usual structure, based on coƯ ordinates, of the elements of analytic geometry, differential and integral calculus and of the theory of differential equations. We are indebted to H. Keller, T. Klemola, T. Nieminen, Ph. Tondeur and K. 1. Virtanen, who read our presentation in our first manuscript, for important critical remarks. The present new English edition deviates at several points from the first edition (d. Introduction). Professor I.S. Louhivaara has from the beginning to the end taken part in the production of the new edition and has advanced our work by suggestions on both content and form. For his important support we wish to express our hearty thanks. We are indebted also to W. Greub and to H. Haahti for various valuable remarks. Our manuscript for this new edition has been translated into English by Doctor P. Emig. We express to him our gratitude for his careful interest and skillful attention during this work In this work an attempt is made to present a systematic basis for a general, absolute, coordinate and dimension free infinitesimal vector calculus. The beginnings for such a calculus appear in the literature quite early. Above all, we should mention the works of M. Frechet, in which the notion of a differential was introduced in a function space. This same trend, the translation of differential calculus to functional analysis, is pursued in a number of later investigations (Gateaux, Hildebrandt, Fischer, Graves, Keller, Kerner, Michal, Elconin, Taylor, Rothe, Sebastiao e Silva, Laugwitz, Bartle, Whitney, Fischer and others). In all of these less attention was paid to classical analysis, the theory of finite dimensional spaces. And yet it seems that already here the absolute point of view offers essential advantages. The elimination of coordinates signifies a gain not only in a formal sense. It leads to a greater unity and simplicity in the theory of functions of arbitrarily many variables, the algebraic structure of analysis is clarified, and at the same time the geometric aspects of linear algebra become more prominent, which simplifies one's ability to comprehend the overall structures and promotes the formation of new ideas and methods Cover Title Foreword Introduction I. Linear Algebra 1. The Linear Space with Real Multiplier Domain 2. Finite Dimensional Linear Spaces 3. Linear Mappings 4. Bilinear and Quadratic Functions 5. Multilincar Functions 6. Metrization of affine Space II. Differential Calculus 1. Derivatives and Differential 2. Taylor's Formula 3. Partial Differentiation 4. Implicit Functions III. Integral Calculus 1. The Affine Integral 2. Theorem of Stokes 3. Applications of Stoke's Theorem IV. Differential Equations 1. Normal System 2. The General Differential Equation of First Order 3. The Linear Differential Equation of Order One V. Theory of Curves and Surfaces 1. Regular Curves and Surfacer 2. Curve Theory 3. Surface Theory 4. Vectors and Tensor 5 Integration of the Derivative Formulas 6. Thcorcma Egregium 7. Parallel Translation 8. The Gauss-Bonnet Theorem VI. Riemannian Geometry 1. Affine Differential Geommtry 2. Riemannian Geometry Bibliography Index