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.

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.

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 Elements of Vector Analysis and Field Theory.- 15 Integration of Differential Forms on Manifolds.- 16 Uniform Convergence and Basic Operations of Analysis.- 17 Integrals Depending on a Parameter.- 18 Fourier Series and the Fourier Transform.- 19 Asymptotic Expansions.- Topics and Questions for Midterm Examinations.- Examination Topics.- Examination Problems (Series and Integrals Depending on a Parameter).- Intermediate Problems (Integral Calculus of Several Variables).- Appendices: A Series as a Tool (Introductory Lecture).- B Change of Variables in Multiple Integrals.- C Multidimensional Geometry and Functions of a Very Large Number of Variables.- D Operators of Field Theory in Curvilinear Coordinates.- E Modern Formula of Newton-Leibniz.- References.- Index of Basic Notation.- Subject Index.- Name Index.