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This secondEnglish edition of a very popular two-volume work presents a thorough firstcourse in analysis, leading from real numbers to such advanced topics asdifferential forms on manifolds; asymptotic methods; Fourier, Laplace, andLegendre transforms; elliptic functions; and distributions. Especially notablein this course are the clearly expressed orientation toward the naturalsciences and the informal exploration of the essence and the roots of the basicconcepts and theorems of calculus. Clarity of exposition is matched by a wealthof instructive exercises, problems, and fresh applications to areas seldomtouched on in textbooks on real analysis.
The maindifference between the second and first English editions is the addition of aseries of appendices to each volume. There are six of them in the first volumeand five in the second. The subjects of these appendices are diverse. They aremeant 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 importantconceptual connections between analysis and other parts of mathematics.This second volumepresents classical analysis in its current form as part of a unifiedmathematics. It shows how analysis interacts with other modern fields ofmathematics such as algebra, differential geometry, differential equations, complex analysis, and functional analysis. This book provides a firm foundationfor advanced work in any of these directions.
Inhaltsverzeichnis
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.
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