Kristian P Evans,Niels Jacob

A Course in Analysis

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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.
Contents:Part 3: Differentiation of Functions of Several Variables:Metric SpacesConvergence and Continuity in Metric SpacesMore on Metric Spaces and Continuous FunctionsContinuous Mappings Between Subsets of Euclidean SpacesPartial DerivativesThe Differential of a MappingCurves in ℝnSurfaces in ℝ3. A First EncounterTaylor Formula and Local Extreme ValuesImplicit Functions and the Inverse Mapping TheoremFurther Applications of the DerivativesCurvilinear CoordinatesConvex Sets and Convex Functions in ℝnSpaces of Continuous Functions as Banach SpacesLine IntegralsPart 4: Integration of Functions of Several Variables:Towards Volume Integrals in the Sense of RiemannParameter Dependent and Iterated IntegralsVolume Integrals on Hyper-RectanglesBoundaries in ℝn and Jordan Measurable SetsVolume Integrals on Bounded Jordan Measurable SetsThe Transformation Theorem : Result and ApplicationsImproper and Parameter Dependent IntegralsPart 5: Vector Calculus:The Scope of Vector CalculusThe Area of a Surface in ℝ3 and Surface IntegralsGauss' Theorem in ℝ3Stokes Theorem in ℝ2 and ℝ3Gauss' Theorem for Domains in ℝnAppendix I : Vector Spaces and Linear MappingsAppendix II : Two Postponed Proofs of Part 3Solutions to Problems of Part 3Solutions to Problems of Part 4Solutions to Problems of Part 5ReferencesMathematicians Contributing to Analysis (Continued)Subject IndexReadership: Undergraduate students in mathematics. Key Features:The course will provide a unique companion for the entire analysis educationFair balance of absolute core and optional materialLarge number of problems solved in great detailThe first author has more than 35 years experience teaching the subject
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