Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 INTRODUCTION TO THE MECHANICAL UNIVERSE (Program 1)
- Chapter 2 THE LAW OF FALLING BODIES (Program 2)
- Chapter 3 THE LANGUAGE OF NATURE: DERIVATIVES AND INTEGRALS
- Chapter 4 INERTIA
- Chapter 5 VECTORS
- Chapter 6 NEWTON'S LAWS AND EQUILIBRIUM
- Chapter 7 UNIVERSAL GRAVITATION AND CIRCULAR MOTION
- Chapter 8 FORCES
- Chapter 9 FORCES IN ACCELERATING REFERENCE FRAMES
- Chapter 10 ENERGY: CONSERVATION AND CONVERSION
- Chapter 11 THE CONSERVATION OF MOMENTUM
- Chapter 12 OSCILLATORY MOTION
- Chapter 13 ANGULAR MOMENTUM
- Chapter 14 ROTATIONAL DYNAMICS FOR RIGID BODIES
- Chapter 15 GYROSCOPES
- Chapter 16 KEPLER'S LAWS AND THE CONIC SECTIONS
- Chapter 17 SOLVING THE KEPLER PROBLEM
- Chapter 18 NAVIGATING IN SPACE
- Chapter 19 TEMPERATURE AND THE GAS LAWS
- Chapter 20 THE ENGINE OF NATURE
- Chapter 21 ENTROPY
- Chapter 22 THE QUEST FOR LOW TEMPERATURE
- Appendix A THE INTERNATIONAL SYSTEM OF UNITS
- Appendix B CONVERSION FACTORS
- Appendix C FORMULAS FROM ALGEBRA, GEOMETRY, AND TRIGONOMETRY
- Appendix D ASTRONOMICAL DATA
- Appendix E PHYSICAL CONSTANTS
- SELECTED BIBLIOGRAPHY
- Index
Chapter 3 - THE LANGUAGE OF NATURE: DERIVATIVES AND INTEGRALS
Published online by Cambridge University Press: 05 August 2013
- Frontmatter
- Contents
- Preface
- Chapter 1 INTRODUCTION TO THE MECHANICAL UNIVERSE (Program 1)
- Chapter 2 THE LAW OF FALLING BODIES (Program 2)
- Chapter 3 THE LANGUAGE OF NATURE: DERIVATIVES AND INTEGRALS
- Chapter 4 INERTIA
- Chapter 5 VECTORS
- Chapter 6 NEWTON'S LAWS AND EQUILIBRIUM
- Chapter 7 UNIVERSAL GRAVITATION AND CIRCULAR MOTION
- Chapter 8 FORCES
- Chapter 9 FORCES IN ACCELERATING REFERENCE FRAMES
- Chapter 10 ENERGY: CONSERVATION AND CONVERSION
- Chapter 11 THE CONSERVATION OF MOMENTUM
- Chapter 12 OSCILLATORY MOTION
- Chapter 13 ANGULAR MOMENTUM
- Chapter 14 ROTATIONAL DYNAMICS FOR RIGID BODIES
- Chapter 15 GYROSCOPES
- Chapter 16 KEPLER'S LAWS AND THE CONIC SECTIONS
- Chapter 17 SOLVING THE KEPLER PROBLEM
- Chapter 18 NAVIGATING IN SPACE
- Chapter 19 TEMPERATURE AND THE GAS LAWS
- Chapter 20 THE ENGINE OF NATURE
- Chapter 21 ENTROPY
- Chapter 22 THE QUEST FOR LOW TEMPERATURE
- Appendix A THE INTERNATIONAL SYSTEM OF UNITS
- Appendix B CONVERSION FACTORS
- Appendix C FORMULAS FROM ALGEBRA, GEOMETRY, AND TRIGONOMETRY
- Appendix D ASTRONOMICAL DATA
- Appendix E PHYSICAL CONSTANTS
- SELECTED BIBLIOGRAPHY
- Index
Summary
It is most useful that the true origins of memorable inventions be known, especially of those which were conceived not by accident but by an effort of meditation. … One of the noblest inventions of our time has been a new kind of mathematical analysis, known as the differential calculus.
Gottfried Wilhelm Leibniz, Historia et origo calculi differentialis (1714)THE DEVELOPMENT OF DIFFERENTIAL CALCULUS
After the advent of algebra in the sixteenth century, a flood of mathematical discoveries swept through Europe. The most important were differential calculus and integral calculus, bold new methods for attacking a host of problems that had challenged the world's best minds for more than 2000 years. Differential calculus deals with ideas such as speed, rate of growth, tangent lines, and curvature, whereas integral calculus treats topics such as area, volume, arc length, and centroids.
Work begun by Archimedes in the third century b.c. led ultimately to the birth of integral calculus in the seventeenth century a.d. This development has a long and fascinating history which we will explore in more detail later.
Differential calculus has a relatively short history. The concept of derivative was first formulated early in the seventeenth century when the French mathematician Pierre de Fermat tried to devise a way of finding the smallest and largest values of a given function. He imagined the graph of a function having, at each of its points, a direction given by a tangent line, as suggested by the points labeled in Fig. 3.1.
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- The Mechanical UniverseMechanics and Heat, Advanced Edition, pp. 27 - 56Publisher: Cambridge University PressPrint publication year: 1986