Mariano Giaquinta, Giuseppe Modica

Mathematical Analysis (eBook, PDF)

Functions of One Variable

• Format: PDF

Produktbeschreibung

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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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 353
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461200079
• Artikelnr.: 44199331

Inhaltsangabe

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.

Rezensionen

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)