Autorentext

Jerrold E. Marsden; Michael J. Hoffman

Klappentext

Designed for courses in advanced calculus and introductory real analysis, the second edition of Elementary Classical Analysis strikes a careful and thoughtful balance between pure and applied mathematics, with the emphasis on techniques important to classical analysis, without vector calculus or complex analysis. As such, it's a perfect teaching and learning resource for mathematics undergraduate courses in classical 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. As well as being suitable for students taking pure mathematics, it can also be used by students taking engineering and physical science courses. There's now even more material on variable calculus, expanding the textbook's already considerable coverage of the subject.

Inhalt

**1. Introduction: Sets and Functions
** Supplement on the Axioms of Set Theory

**2. 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

**3. Topology of Euclidean Space
** Open Sets
Interior of a Set
Closed Sets
Accumulation Points
Closure of a Set
Boundary of a Set
Sequences
Completeness
Series of Real Numbers and Vectors

**4. Compact and Connected Sets
** Compacted-ness
The Heine-Borel Theorem
Nested Set Property
Path-Connected Sets
Connected Sets

**5. Continuous Mappings
** Continuity
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

**6. 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

**7. 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
Taylors Theorem and Higher Derivatives
Maxima and Minima

**8. 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

**9. Integration
** Integrable Functions
Volume and Sets of Measure Zero
Lebesgues Theorem
Properties of the Integral
Improper Integrals
Some Convergence Theorems
Introduction to Distributions

**10. Fubinis Theorem and the Change of Variables Formula
** Introduction
Fubinis Theorem
Change of Variables Theorem
Polar Coordinates
Spherical Coordinates and Cylindrical Coordinates
A Note on the Lebesgue Integral
Interchange of Limiting Operations

**11. Fourier Analysis
** Inner Product Spaces
Orthogonal Families of Functions
Completeness and Convergence Theorems
Functions of Bounded Variation and Fejér Theory (Optional)
Computation of Fourier Series
Further Convergence Theorems
Applications
Fourier Integrals
Quantum Mechanical Formalism

**Miscellaneous Exercises
** **References
** **Answers to Selected Odd-Numbered Exercises
** Index