Practical Analysis in One Variable

Donald Estep

622 pages, parution le 31/10/2002

Ajouter à une liste

Indisponible

Résumé

This book attempts to place the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to young students. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. The book can be used in an honor calculus sequence typically taken by freshmen planning to major in engineering, mathematics, and science, or in an introductory course in rigorous real analysis offered to mathematics majors.
Students who work through the proofs and solve the practical problems in this book will develop a "hands-on" understanding of analysis that will serve them well in the future.

Contents

Introduction
I. Numbers and Functions, Sequences and Limits
Mathematical Modeling
Natural Numbers Just Aren't Enough
Infinity and Mathematical Induction
Rational Numbers
Functions
Polynomials
Functions, Functions, and More Functions
Lipschitz Continuity
Sequences and Limits
Solving the Muddy Yard Model
Real Numbers
Functions of Real Numbers
The Bisection Algorithm
Inverse Functions
Fixed Points and Contraction Maps
II. Differential and Integral Calculus
The Linearization of a Function at a Point
Analyzing the Behavior of a Population Model
Interpretations of the Derivative
Differentiability on Intervals
Useful Properties of the Derivative
The Mean Value Theorem
Derivatives of Inverse Functions
Modeling with Differential Equations
Antidifferentiation
Integration.Properties of the Integral
Applications of the Integral
Rocket Propulsion and the Logarithm
Constant Relative Rate of Change and the Exponential
A Mass-Spring System and the Trigonometric Functions
Fixed Point Iteration and Newton's Method
Calculus Quagmires
III. You Want Analysis? We've Got Your Analysis Right Here
Notions of Continuity and Differentiability
Sequences of Functions
Relaxing Integration
Delicate Limits and Gross Behavior
The Weierstrass Approximation Theorem
The Taylor Polynomial
Polynomial Interpolation
Nonlinear Differential Equations
The Picard Iteration
The Forward Euler Method
A Conclusion or a Beginning?
References
Index

L'auteur - Donald Estep

Estep, Donald Colorado State University, Fort Collins, CO, USA

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Donald Estep
Parution	31/10/2002
Nb. de pages	622
Format	16 x 24
Couverture	Relié
Poids	1055g
Intérieur	Noir et Blanc
EAN13	9780387954844
ISBN13	978-0-387-95484-4