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Full Description
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.
Contents
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.
