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Mathematical Analysis: Functions of One Variable de Mariano Giaquinta

Mathematical Analysis: Functions of One Variable

Autor Mariano Giaquinta, Giuseppe Modica
en Limba Engleză Hardback – 15 mai 2003
For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. Unfortunately, professional representatives of mathematics share in the reponsibiIity. The teaching of mathematics has sometimes degen­ erated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual indepen­ dence. Mathematical research has shown a tendency toward overspecialization and over-emphasis on abstraction. Applications and connections with other fields have been neglected . . . But . . . understanding of mathematics cannot be transmitted by painless entertainment any more than education in music can be brought by the most brilliant journalism to those who never have lis­ tened intensively. Actual contact with the content of living mathematics is necessary. Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism which refuses to disclose motive or goal and which is an unfair obstacle to honest effort. (From the preface to the first edition of What is Mathematics? by Richard Courant and Herbert Robbins, 1941.
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Specificații

ISBN-13: 9780817643126
ISBN-10: 0817643125
Pagini: 353
Ilustrații: XIII, 353 p.
Dimensiuni: 155 x 235 x 22 mm
Greutate: 0.64 kg
Ediția:2003
Editura: Birkhäuser Boston
Colecția Birkhäuser
Locul publicării:Boston, MA, United States

Public țintă

Graduate

Cuprins

1. Numbers, Functions and their Graphs.- 1.1 Real Numbers: a Description.- 1.2 The Cartesian Plane.- 1.3 Elementary Functions.- 1.4 Remarks on Common Language and the Language of Mathematics.- 1.5 Exercises.- 2. Limits and Continuity.- 2.1 Limits.- 2.2. Continuous Functions.- 2.3. Continuous functions on an interval.- 2.4 Weierstrass’s Theorem.- 2.5 Summing Up.- 2.6 Exercises.- 3. The Fundamental Ideas of the Differential and Integral Calculus.- 3.1 Differential Calculus.- 3.2 Integral Calculus.- 3.3 The Fundamental Theorem of Calculus.- 3.4 Calculus: Some Historical Remarks.- 3.5 Summing Up.- 3.6 Exercises.- 4. The Calculus of Derivatives and of Integrals.- 4.1 Computation of Derivatives.- 4.2 Integrals and Primitives.- 4.3 A Definition of the Trigonometric, Logarithmic and Exponential Functions.- 4.4 Some Differential Equations.- 4.5 Generalized Integrals.- 4.6 Summing Up.- 4.7 Exercises.- 5. Further Developments in Calculus.- 5.1 Taylor’s Formula.- 5.2 The Calculus of Limits.- 5.3 Convex Functions.- 5.4 Some Inequalities.- 5.5 Graphing a Function.- 5.6 Summing Up.- 5.7 Exercises.- 6. Toward Differential Equations and Minimum Principles.- 6.1 Linear Ordinary Differential Equations.- 6.2 First Order ODEs.- 6.3 One-Dimensional Motions.- 6.4 Optimization Problems.- 6.5 Summing Up.- 6.6 Exercises.- A. Matematicians and Other Scientists.- B. Bibliographical Notes.- C. Index.

Recenzii

From the reviews:
"Many teachers and students could benefit from thinking about its unique perspectives."
---SIAM Review
"Giaquinta . . . and Modica . . . present the differential and integral calculus for real functions of a single real variable from a sophisticated point of view. Their book is far better suited as a source of unique perspectives on the logical development of calculus ideas and their physical applications . . . [so that] it will better serve the experienced calculus instructor seeking new approaches to familiar material, or the student who has already mastered the basic ideas and techniques of calculus but wishes to see it all again, 'done right,' . . . Highly recommended."
---CHOICE
"This self-contained book introduces the main ideas and fundamental methods of mathematical analysis without loosing sight of the context in which it was developed and the role played in science, particularly in physics…. Each chapter has a short summary where the most important facts discussed are collected. There is also a large number of exercises inserted at various points into the text…. The book is meant principally for students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics."
---ZENTRALBLATT MATH
"The presentation of the theory is clearly arranged, all theorems have rigorous proofs and every chapter closes with a summing up of the results and exericeses with different requirements. The aim of the authors was very successful[ly] realized, and this book is excellently suitable for students in mathematics, physics, engineering, computer science and all students of technological and scientific faculties." ---Journal of Analysis and its Applications
"This is the first of several volumes on analysis by theseauthors. … It is essentially a calculus book at the advanced undergraduate level, or an analysis textbook for someone who has never seen calculus before. … I like this book … ." (Warren Johnson, The Mathematical Association of America, October, 2009)

Caracteristici

Interesting and valuable historical account of ideas and methods in analysis with beautiful illustrations Rigorous exposition with full proofs motivated by numerous examples Exercises, comprehensive bibliography and index

Textul de pe ultima copertă

Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables builds upon the basic ideas and techniques of differential and integral calculus for functions of several variables, as outlined in an earlier introductory volume. The presentation is largely focused on the foundations of measure and integration theory.
The book begins with a discussion of the geometry of Hilbert spaces, convex functions and domains, and differential forms, particularly k-forms. The exposition continues with an introduction to the calculus of variations with applications to geometric optics and mechanics. The authors conclude with the study of measure and integration theory – Borel, Radon, and Hausdorff measures and the derivation of measures. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.
This work may be used as a supplementary text in the classroom or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering. One of the key strengths of this presentation, along with the other four books on analysis published by the authors, is the motivation for understanding the subject through examples, observations, exercises, and illustrations.
Other books published by the authors – all of which provide the reader with a strong foundation in modern-day analysis – include:
* Mathematical Analysis: Functions of One Variable
* Mathematical Analysis: Approximation and Discrete Processes
* Mathematical Analysis: Linear and Metric Structures and Continuity
* Mathematical Analysis: An Introduction to Functions of Several Variables
Reviews of previous volumes of Mathematical Analysis:
The presentation of the theory is clearly arranged, all theorems haverigorous proofs, and every chapter closes with a summing up of the results and exercises with different requirements. . . . This book is excellently suitable for students in mathematics, physics, engineering, computer science and all students of technological and scientific faculties.
—Journal of Analysis and its Applications
The exposition requires only a sound knowledge of calculus and the functions of one variable.  A key feature of this lively yet rigorous and systematic treatment is the historical accounts of ideas and methods of the subject.   Ideas in mathematics develop in cultural, historical and economical contexts, thus the authors made brief accounts of those aspects and used a large number of beautiful illustrations.  
—Zentralblatt MATH