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. 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 The book begins at an undergraduate student level, assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, the Lebesgue integral, vector calculus and differential equations. After having created 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 mathematicians today.
