Calculus
Description
When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus. Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.
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Otto Toeplitz (1881-1940) was a leading scholar in linear algebra and functional analysis. He was the author of many scholarly articles and coauthor of The Enjoyment of Mathematics.
Content
- Intro
- TABLE OF CONTENTS
- FOREWORD
- PREFACE TO THE GERMAN EDITION
- PREFACE TO THE ENGLISH EDITION
- I. THE NATURE OF THE INFINITE PROCESS
- 1. The Beginnings of Greek Speculation on Infinitesimals
- 2. The Greek Theory of Proportions
- 3. The Exhaustion Method of the Greeks
- 4. The Modern Number Concept
- 5. Archimedes' Measurements of the Circle and the Sine Tables
- 6. The Infinite Geometric Series
- 7. Continuous Compound Interest
- 8. Periodic Decimal Fractions
- 9. Convergence and Limit
- 10. Infinite Series
- II. THE DEFINITE INTEGRAL
- 11. The Quadrature of the Parabola by Archimedes
- 12. Continuation after 1,880 Years
- 13. Area and Definite Integral
- 14. Non-rigorous Infinitesimal Methods
- 15. The Concept of the Definite Integral
- 16. Some Theorems on Definite Integrals
- 17. Questions of Principle
- III. DIFFERENTIAL AND INTEGRAL CALCULUS
- 18. Tangent Problems
- 19. Inverse Tangent Problems
- 20. Maxima and Minima
- 21. Velocity
- 22. Napier
- 23. The Fundamental Theorem
- 24. The Product Rule
- 25. Integration by Parts
- 26. Functions of Functions
- 27. Transformation of Integrals
- 28. The Inverse Function
- 29. Trigonometric Functions
- 30. Inverse Trigonometric Functions
- 31. Functions of Several Functions
- 32. Integration of Rational Functions
- 33. Integration of Trigonometric Expressions
- 34. Integration of Expressions Involving Radicals
- 35. Limitations of Explicit Integration
- IV. APPLICATIONS TO PROBLEMS OF MOTION
- 36. Velocity and Acceleration
- 37. The Pendulum
- 38. Coordinate Transformations
- 39. Elastic Vibrations
- 40. Kepler's First Two Laws
- 41. Derivation of Kepler's First Two Laws from Newton's Law
- 42. Kepler's Third Law
- EXERCISES
- BIBLIOGRAPHY
- Works on the History of Mathematics
- Special Works on the History of the Infinitesimal Calculus
- BIBLIOGRAPHICAL NOTES
- INDEX
