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June 28, 2025, 3:56 p.m.
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SOURCE: Jacob N. Course in Analysis Vol II. Differentiation and Integration...2016
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COVER
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MEDIAINFO
Textbook in PDF format This is the second volume of "A Course in Analysis" and it is devoted to the study of mappings between subsets of Euclidean spaces. The metric, hence the topological structure is discussed as well as the continuity of mappings. This is followed by introducing partial derivatives of real-valued functions and the differential of mappings. Many chapters deal with applications, in particular to geometry (parametric curves and surfaces, convexity), but topics such as extreme values and Lagrange multipliers, or curvilinear coordinates are considered too. On the more abstract side results such as the Stone–Weierstrass theorem or the Arzela–Ascoli theorem are proved in detail. The first part ends with a rigorous treatment of line integrals. The second part handles iterated and volume integrals for real-valued functions. Here we develop the Riemann (–Darboux–Jordan) theory. A whole chapter is devoted to boundaries and Jordan measurability of domains. We also handle in detail improper integrals and give some of their applications. The final part of this volume takes up a first discussion of vector calculus. Here we present a working mathematician's version of Green's, Gauss' and Stokes' theorem. Again some emphasis is given to applications, for example to the study of partial differential equations. At the same time we prepare the student to understand why these theorems and related objects such as surface integrals demand a much more advanced theory which we will develop in later volumes. This volume offers more than 260 problems solved in complete detail which should be of great benefit to every serious student. Preface Introduction List of Symbols Differentiation of Functions of Several Variables Metric Spaces Convergence and Continuity in Metric Spaces More on Metric Spaces and Continuous Functions Continuous Mappings Between Subsets of Euclidean Spaces Partial Derivatives The Differential of a Mapping Curves in ℝn Surfaces in ℝ3. A First Encounter Taylor Formula and Local Extreme Values Implicit Functions and the Inverse Mapping Theorem Further Applications of the Derivatives Curvilinear Coordinates Convex Sets and Convex Functions in ℝn Spaces of Continuous Functions as Banach Spaces Line Integrals Integration of Functions of Several Variables Towards Volume Integrals in the Sense of Riemann Parameter Dependent and Iterated Integrals Volume Integrals on Hyper-Rectangles Boundaries in ℝn and Jordan Measurable Sets Volume Integrals on Bounded Jordan Measurable Sets The Transformation Theorem: Result and Applications Improper Integrals and Parameter Dependent Integrals Vector Calculus The Scope of Vector Calculus The Area of a Surface in ℝ3 and Surface Integrals Gauss’ Theorem in ℝ3 Stokes’ Theorem in ℝ2 and ℝ3 Gauss’ Theorem for ℝn Appendices Vector Spaces and Linear Mappings Two Postponed Proofs of Part 3 Solutions to Problems of Part 3 Solutions to Problems of Part 4 Solutions to Problems of Part 5 Mathematicians Contributing to Analysis (Continued) Subject Index
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