Mathematical Analysis A Concise Introduction

ISBN-10: 0470107960
ISBN-13: 9780470107966
Edition: 2008
List price: $128.00 Buy it from $42.99
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Description: Bridging the gap between calculus and further abstract topics this book presents a well organized and much needed introduction to the foundations of analysis. It is composed of three sections: the analysis of functions of one real variable,  More...

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Book details

List price: $128.00
Copyright year: 2008
Publisher: John Wiley & Sons, Incorporated
Publication date: 11/12/2007
Binding: Hardcover
Pages: 584
Size: 6.50" wide x 9.50" long x 1.50" tall
Weight: 2.310

Bridging the gap between calculus and further abstract topics this book presents a well organized and much needed introduction to the foundations of analysis. It is composed of three sections: the analysis of functions of one real variable, including an introduction to the Lebesgue integral; how the appropriate abstractions lead to a powerful and widely applicable theoretical foundation for all branches of applied mathematics; an outlook to applied subjects in which analysis is used.

Table of Contents
Preface
Analysis of Functions of a Single Real Variable
The Real Numbers
Field Axioms
Order Axioms
Lowest Upper and Greatest Lower Bounds
Natural Numbers, Integers, and Rational Numbers
Recursion, Induction, Summations, and Products
Sequences of Real Numbers
Limits
Limit Laws
Cauchy Sequences
Bounded Sequences
Infinite Limits
Continuous Functions
Limits of Functions
Limit Laws
One-Sided Limits and Infinite Limits
Continuity
Properties of Continuous Functions
Limits at Infinity
Differentiable Functions
Differentiability
Differentiation Rules
Rolle's Theorem and the Mean Value Theorem
The Riemann Integral I
Riemann Sums and the Integral
Uniform Continuity and Integrability of Continuous Functions
The Fundamental Theorem of Calculus
The Darboux Integral
Series of Real Numbers I
Series as a Vehicle To Define Infinite Sums
Absolute Convergence and Unconditional Convergence
Some Set Theory
The Algebra of Sets
Countable Sets
Uncountable Sets
The Riemann Integral II
Outer Lebesgue Measure
Lebesgue's Criterion for Riemann Integrability
More Integral Theorems
Improper Riemann Integrals
The Lebesgue Integral
Lebesgue Measurable Sets
Lebesgue Measurable Functions
Lebesgue Integration
Lebesgue Integrals versus Riemann Integrals
Series of Real Numbers II
Limits Superior and Inferior
The Root Test and the Ratio Test
Power Series
Sequences of Functions
Notions of Convergence
Uniform Convergence
Transcendental Functions
The Exponential Function
Sine and Cosine
L'Hopital's Rule
Numerical Methods
Approximation with Taylor Polynomials
Newton's Method
Numerical Integration
Analysis in Abstract Spaces
Integration on Measure Spaces
Measure Spaces
Outer Measures
Measurable Functions
Integration of Measurable Functions
Monotone and Dominated Convergence
Convergence in Mean, in Measure, and Almost Everywhere
Product [sigma]-Algebras
Product Measures and Fubini's Theorem
The Abstract Venues for Analysis
Abstraction I: Vector Spaces
Representation of Elements: Bases and Dimension
Identification of Spaces: Isomorphism
Abstraction II: Inner Product Spaces
Nicer Representations: Orthonormal Sets
Abstraction III: Normed Spaces
Abstraction IV: Metric Spaces
L[superscript p] Spaces
Another Number Field: Complex Numbers
The Topology of Metric Spaces
Convergence of Sequences
Completeness
Continuous Functions
Open and Closed Sets
Compactness
The Normed Topology of R[superscript d]
Dense Subspaces
Connectedness
Locally Compact Spaces
Differentiation in Normed Spaces
Continuous Linear Functions
Matrix Representation of Linear Functions
Differentiability
The Mean Value Theorem
How Partial Derivatives Fit In
Multilinear Functions (Tensors)
Higher Derivatives
The Implicit Function Theorem
Measure, Topology, and Differentiation
Lebesgue Measurable Sets in R[superscript d]
C[infinity] and Approximation of Integrable Functions
Tensor Algebra and Determinants
Multidimensional Substitution
Introduction to Differential Geometry
Manifolds
Tangent Spaces and Differentiable Functions
Differential Forms, Integrals Over the Unit Cube
k-Forms and Integrals Over k-Chains
Integration on Manifolds
Stokes' Theorem
Hilbert Spaces
Orthonormal Bases
Fourier Series
The Riesz Representation Theorem
Applied Analysis
Physics Background
Harmonic Oscillators
Heat and Diffusion
Separation of Variables, Fourier Series, and Ordinary Differential Equations
Maxwell's Equations
The Navier Stokes Equation for the Conservation of Mass
Ordinary Differential Equations
Banach Space Valued Differential Equations
An Existence and Uniqueness Theorem
Linear Differential Equations
The Finite Element Method
Ritz-Galerkin Approximation
Weakly Differentiable Functions
Sobolev Spaces
Elliptic Differential Operators
Finite Elements
Conclusion and Outlook
Appendices
Logic
Statements
Negations
Set Theory
The Zermelo-Fraenkel Axioms
Relations and Functions
Natural Numbers, Integers, and Rational Numbers
The Natural Numbers
The Integers
The Rational Numbers
Bibliography
Index

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