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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.
Part 3: Differentiation of Functions of Several Variables:
Convergence and Continuity in Metric Spaces
More on Metric Spaces and Continuous Functions
Continuous Mappings Between Subsets of Euclidean Spaces
The Differential of a Mapping
Curves in ?n
Surfaces in ?3. A First Encounter
Taylor Formula and Local Extreme Values
Implicit Functions and the Inverse Mapping Theorem
Further Applications of the Derivatives
Convex Sets and Convex Functions in ?n
Spaces of Continuous Functions as Banach Spaces
Part 4: 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 ?n and Jordan Measurable Sets
Volume Integrals on Bounded Jordan Measurable Sets
The Transformation Theorem : Result and Applications
Improper and Parameter Dependent Integrals
Part 5: Vector Calculus:
The Scope of Vector Calculus
The Area of a Surface in ?3 and Surface Integrals
Gauss' Theorem in ?3
Stokes Theorem in ?2 and ?3
Gauss' Theorem for Domains in ?n
Appendix I : Vector Spaces and Linear Mappings
Appendix II : 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)
Readership: Undergraduate students in mathematics.
The course will provide a unique companion for the entire analysis education
Fair balance of absolute core and optional material
Large number of problems solved in great detail
The first author has more than 35 years experience teaching the subject