Mathematical Analysis II

Front Cover
Springer, Feb 12, 2016 - Mathematics - 720 pages

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.

This second volume presents classical analysis in its current form as part of a unified mathematics. It shows how analysis interacts with other modern fields of mathematics such as algebra, differential geometry, differential equations, complex analysis, and functional analysis. This book provides a firm foundation for advanced work in any of these directions.

 

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Contents

Continuous Mappings General Theory
1
Differential Calculus from a More General Point of View
41
Multiple Integrals
109
Surfaces and Differential Forms in Rn
163
Line and Surface Integrals
213
Elements of Vector Analysis and Field Theory
256
Integration of Differential Forms on Manifolds
313
Uniform Convergence and the Basic Operations of Analysis on Series and Families of Functions
363
Examination Problems Series and Integrals Depending on a Parameter
643
Intermediate Problems Integral Calculus of Several Variables
644
Series as a Tool Introductory Lecture
645
Change of Variables in Multiple Integrals Deduction and First Discussion of the Change of Variables Formula1
655
Multidimensional Geometry and Functions of a Very Large Number of Variables Concentration of Measures and Laws of Large Numbers
663
Operators of Field Theory in Curvilinear Coordinates
670
Modern Formula of NewtonLeibniz and the Unity of Mathematics Final Survey
683
References
693

Integrals Depending on a Parameter
405
Fourier Series and the Fourier Transform
493
Asymptotic Expansions
587
Topics and Questions for Midterm Examinations
633
Examination Topics
638

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About the author (2016)

VLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and mathematical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings. He holds a patent in the technology of mechanical engineering, and he is also known by his book Mathematical Analysis of Problems in the Natural Sciences.

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