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Elementary Classical Analysis - Jerrold Marsden

Elementary Classical Analysis


Published: 15th March 2014
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Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. The book includes detailed coverage of the foundations of the real number system and focuses primarily on analysis in Euclidean space with a view towards application. Intended for students of engineering and physical science as well as of pure mathematics.

Introduction: Sets and Functions
Supplement on the Axioms of Set Theory
The Real Line and Euclidean Space
Ordered Fields and the Number Systems
Completeness and the Real Number System
Least Upper Bounds
Cauchy Sequences
Cluster Points: lim inf and lim sup
Euclidean Space
Norms, Inner Products, and Metrics
The Complex Numbers
Topology of Euclidean Space
Open Sets
Interior of a Set
Closed Sets
Accumulation Points
Closure of a Set
Boundary of a Set
Series of Real Numbers and Vectors
Compact and Connected Sets
The Heine-Borel Theorem
Nested Set Property
Path-Connected Sets
Connected Sets
Continuous Mappings
Images of Compact and Connected Sets
Operations on Continuous Mappings
The Boundedness of Continuous Functions of Compact Sets
The Intermediate Value Theorem
Uniform Continuity
Differentiation of Functions of One Variable
Integration of Functions of One Variable
Uniform Convergence
Pointwise and Uniform Convergence
The Weierstrass M Test
Integration and Differentiation of Series
The Elementary Functions
The Space of Continuous Functions
The Arzela-Ascoli Theorem
The Contraction Mapping Principle and Its Applications
The Stone-Weierstrass Theorem
The Dirichlet and Abel Tests
Power Series and Cesaro and Abel Summability
Differentiable Mappings
Definition of the Derivative
Matrix Representation
Continuity of Differentiable Mappings; Differentiable Paths
Conditions for Differentiability
The Chain Rule
Product Rule and Gradients
The Mean Value Theorem
Taylor's Theorem and Higher Derivatives
Maxima and Minima
The Inverse and Implicit Function Theorems and Related Topics
Inverse Function Theorem
Implicit Function Theorem
The Domain-Straightening Theorem
Further Consequences of the Implicit Function Theorem
An Existence Theorem for Ordinary Differential Equations
The Morse Lemma
Constrained Extrema and Lagrange Multipliers
Integrable Functions
Volume and Sets of Measure Zero
Lebesgue's Theorem
Properties of the Integral
Improper Integrals
Some Convergence Theorems
Introduction to Distributions
Fubini's Theorem and the Change of Variables Formula
Fubini's Theorem
Change of Variables Theorem
Polar Coordinates
Spherical Coordinates and Cylindrical Coordinates
A Note on the Lebesgue Integral
Interchange of Limiting Operations
Fourier Analysis
Inner Product Spaces
Orthogonal Families of Functions
Completeness and Convergence Theorems
Functions of Bounded Variation and Fejã©r Theory (Optional)
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780716728863
ISBN-10: 0716728869
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 600
Published: 15th March 2014
Publisher: W.H.Freeman & Co Ltd
Country of Publication: US
Dimensions (cm): 24.0 x 16.0
Weight (kg): 0.455
Edition Number: 3
Edition Type: Revised