Mathematical Analysis vol 2

This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, integral transforms, and distributions. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books. The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions. This two-volume work by V.A.Zorich on Mathematical Analysis constitutes a thorough first course in real analysis, leading from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, and elliptic functions. With masterful exposition, the author provides a smooth, gradual transition from each topic to the next, so that the slope never feels too steep for the reader. Making use of Cartan's concept of a filter base, the author disperses the fog of epsilons and deltas that have always made the crucial subject of limits a barrier for the nonmathematical specialist. As a result, the major theorems of differentiation and integrationreveal their essential unity in a nearly painless manner. The clarity of the exposition is matched by a wealth of instructive exercises and fresh applications to areas seldom touched on in real analysis books, many of which are taken from physics and technology. TOC:Prefaces.- 9 Continuous Mappings (General Theory).- 10 Differential Calculus from a General Viewpoint.- 11 Multiple Integrals.- 12 Surfaces and Differential Forms in Rn.- 13 Line and Surface Integrals.- 14 The Elements of Vector Analysis and Field Theory.- 15 Integration of Differential Forms on Manifolds.- 16 Uniform Convergence and the Basic Operations of Analysis.- 17 Integrals Depending on a Parameter.- 18 Fourier Series and the Fourier Transform.- 19 Asymptotic Expansions.- Some Problems from the Midterm Examinations.- Examination Topics.- References.- Subject Index.- Name Index This two-volume work by V.A.Zorich on Mathematical Analysis constitutes a thorough first course in real analysis, leading from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, and elliptic functions. With masterful exposition, the author provides a smooth, gradual transition from each topic to the next, so that the slope never feels too steep for the reader. Making use of Cartan's concept of a filter base, the author disperses the fog of epsilons and deltas that have always made the crucial subject of limits a barrier for the nonmathematical specialist. As a result, the major theorems of differentiation and integrationreveal their essential unity in a nearly painless manner. The clarity of the exposition is matched by a wealth of instructive exercises and fresh applications to areas seldom touched on in real analysis books, many of which are taken from physics and technology. TOC:Prefaces.- 1. Some General Mathematical Concepts and Notation.- 2. The Real Numbers.- 3. Limits.- 4. Continuous Functions.- 5. Differential Calculus.- 6. Integration.- 7. Functions of Several Variables.- 8. Differential Calculus in Several Variables.- Some Problems from the Midterm Examinations.- Examination Topics.- References.- Subject Index.- Name Index

This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, integral transforms, and distributions. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books.

The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.

This Work By Zorich On Mathematical Analysis Constitutes A Thorough First Course In Real Analysis, Leading From The Most Elementary Facts About Real Numbers To Such Advanced Topics As Differential Forms On Manifolds, Asymptotic Methods, Fourier, Laplace, And Legendre Transforms, And Elliptic Functions. V.1. Some General Mathematical Concepts And Notation -- The Real Numbers -- Limits -- Continuous Functions -- Differential Calculus -- Integration -- Functions Of Several Variables -- Differential Calculus In Several Variables -- Some Problems From The Midterm Examinations. Vladimir A. Zorich ; [translator, Roger Cooke]. Includes Bibliographical References And Indexes. FIRST VOLUME OF THE AMAZING 2 VOLUME ANALYSIS TEXT BY THE INTERNATIONALLY REKNOWNED EXPERT,BASED ON COURSES TAUGHT AT MOSCOW STATE UNIVERSITY FOR DECADES TO GIFTED FRESHMAN,UNIQUE IN IT'S APPROACH IN THAT IT COMBINES THE HARD THEORY OF REAL AND COMPLEX ANALYSIS AND APPLICATIONS TO PHYSICS IN THE SAME PRESENTATION. MOST ANALYSIS TEXTS DON'T COVER BOTH AND THIS PRESENTATION IS VERY UNIQUE IN THIS MANNER. THIS IS A NEARLY PERFECT COPY OF THE 2004 1ST HARDCOVER EDITION,A GREAT COPY FOR CLASS! Definition 1. A set X is said to be endowed with a metric or a metric space structure or to be a metric space if a function d : X x X R (9.1) is exhibited satisfying the following conditions: a) d(x1, x2) = 0 x1 = x2, b) d(x1, x2) = d(x2, x2) (symmetry), c) d(x1, x3) d(x1, x2) + d(x2, x3) (the triangle inequality), where x1, x2, x3 are arbitrary elements of X. The language of this book, like the majority of mathematical texts, consists of ordinary language and a number of special symbols from the theories being discussed.