Mathematical Analysis II (Universitext)

Main subject categories: • Mathematical analysis • Differential calculus of one variable • Integral calculus of one variable • Differential calculus of several variables • Integral calculus of several variablesMathematics Subject Classification (2010): 26-01 Introductory exposition pertaining to real functions • 26Axx Functions of one variable • 26Bxx Functions of several variables • 42-01 Introductory exposition pertaining to harmonic analysis on Euclidean spacesThe 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 second English 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; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the 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 textbooks on real analysis. The main difference between the second and first English editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.

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 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, Fourier, Laplace, And Legendre Transforms, Elliptic Functions 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 First Volume Constitutes A Complete Course On One-variable Calculus Along With The Multivariable Differential Calculus Elucidated In An Up-to-day, Clear Manner, With A Pleasant Geometric Flavor.

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. Presents a course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, elliptic functions and distributions. This book explores the essence and the roots of the basic concepts and theorems of calculus. Some general mathematical concepts and notation The real numbers Limits Continuous functions Differential calculus Integration Functions of several variables Differential calculus in several variables.