The book begins at the level of an undergraduate student assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations. After having built on a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis, as understood by a mathematician today.
Author(s): Igor Kriz, Aleš Pultr (auth.)
Edition: 1
Publisher: Birkhäuser Basel
Year: 2013
Language: English
Pages: 510
Tags: Real Functions; Linear and Multilinear Algebras, Matrix Theory; Measure and Integration; Functions of a Complex Variable; Ordinary Differential Equations; Sequences, Series, Summability
Front Matter....Pages i-xx
Front Matter....Pages 1-1
Preliminaries....Pages 3-31
Metric and Topological Spaces I....Pages 33-63
Multivariable Differential Calculus....Pages 65-95
Integration I: Multivariable Riemann Integral and Basic Ideas Toward the Lebesgue Integral....Pages 97-116
Integration II: Measurable Functions, Measure and the Techniques of Lebesgue Integration....Pages 117-143
Systems of Ordinary Differential Equations....Pages 145-173
Systems of Linear Differential Equations....Pages 175-191
Line Integrals and Green’s Theorem....Pages 193-210
Front Matter....Pages 211-211
Metric and Topological Spaces II....Pages 213-235
Complex Analysis I: Basic Concepts....Pages 237-266
Multilinear Algebra....Pages 267-285
Smooth Manifolds, Differential Forms and Stokes’ Theorem....Pages 287-310
Complex Analysis II: Further Topics....Pages 311-348
Calculus of Variations and the Geodesic Equation....Pages 349-366
Tensor Calculus and Riemannian Geometry....Pages 367-392
Banach and Hilbert Spaces: Elements of Functional Analysis....Pages 393-426
A Few Applications of Hilbert Spaces....Pages 427-449
Back Matter....Pages 451-510
