Mathematical Analysis II
by Zorich, Vladimir A.; Cooke, RogerFormat: Paperback
Pub. Date: 2008-12-01
Publisher(s): Springer Nature
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Summary
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.
Table of Contents
| Contents of Volume II | |
| Prefaces | |
| Preface to the fourth edition | |
| Prefact to the third edition | |
| Preface to the second edition | |
| Preface to the first edition | |
| Continuous Mappings (General Theory) | |
| Metric spaces | |
| Definitions and examples | |
| Open and closed subsets of a metric space | |
| Subspaces of a metric space | |
| The direct product of metric spaces | |
| Problems and exercises | |
| Topological spaces | |
| Basic definitions | |
| Subspaces of a topological space | |
| The direct product of topological spaces | |
| Problems and exercises | |
| Compact sets | |
| Definition and general properties of compact sets | |
| Metric compact sets | |
| Problems and exercises | |
| Connected topological spaces | |
| Problems and exercises | |
| Complete metric spaces | |
| Basic definitions and examples | |
| The completion of a metric space | |
| Problems and exercises | |
| Continuous mappings of topological spaces | |
| The limit of a mapping | |
| Continuous mappings | |
| Problems and exercises | |
| The contraction mapping principle | |
| Problems and exercises | |
| *Differential Calculus from a General Viewpoint | |
| Normed vector spaces | |
| Some examples of the vector spaces of analysis | |
| Norms in vector spaces | |
| Inner products in a vector space | |
| Problems and exercises | |
| Linear and multilinear transformations | |
| Definitions and examples | |
| The norm of a transformation | |
| The space of continuous transformations | |
| Problems and exercises | |
| The differential of a mapping | |
| Mappings differentiable at a point | |
| The general rules for differentiation | |
| Some examples | |
| The partial deriatives of a mapping | |
| Problems and exercises | |
| The mean-value theorem and some examples of its use | |
| The mean-value theorem | |
| Some applications of the mean-value theorem | |
| Problems and exercises | |
| Higher-order derivatives | |
| Definition of the nth differential | |
| The derivative with respect to a vector and the computation of the values of the nth differential | |
| Symmetry of the higher-order differentials | |
| Some remarks | |
| Problems and exercises | |
| Taylor's formula and methods of finding extrema | |
| Taylor's formula for mappings | |
| Methods of finding interior extrema | |
| Some examples | |
| Problems and exercises | |
| The general implicit function theorem | |
| Problems and exercises11 Multiple Integrals | |
| The Riemann integral over an n-dimensional interval | |
| Definition of the integral | |
| The Lebesgue criterion for Riemann integrability | |
| The Darboux criterion | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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