Jacob N., Evans K.P. Course in Analysis. Vol. II Differentiation and Integration of Functions of Several Variables Vector Calculus

World Scientific, 2016. — 788 p. — ISBN: 978-981-3140-95-0, 978-981-3140-96-7, 978-981-3140-98-1. 978-981-3140-97-4.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 SymbolsDifferentiation 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 IntegralsIntegration 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 IntegralsVector 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 ℝⁿAppendices
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 5Mathematicians Contributing to Analysis (Continued)
Subject Index