Calculus Gems微积分精华
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作者： George F. Simmons 著
出 版 社：
出版时间： 2007-1-12字数：版次： 1页数： 355印刷时间： 2007/01/12开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780883855614包装： 精装内容简介
A classic book is back in print! It can be used as a supplement in a Calculus course, or a History of Mathematics course. The first half of Calculus Gems entitles, Brief Lives is a biographical history of mathematics from the earliest times to the late nineteenth century. The author shows that Science and mathematics in particular is something that people do, and not merely a mass of observed data and abstract theory. He demonstrates the profound connections that join mathematics to the history of philosophy and also to the broader intellectual and social history of Western civilization. The second half of the book contains nuggets that Simmons has collected from number theory, geometry, science, etc., which he has used in his mathematics classes. G.H. Hardy once said, A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. This part of the book contains a wide variety of these patterns, arranged in an order roughly corresponding to the order of the ideas in most calculus courses. Some of the sections even have a few problems. Professor Simmons tells us in the Preface of Calculus Gems: I hold the naïve but logically impeccable view that there are only two kinds of students in our colleges and universities, those who are attracted to mathematics; and those who are not yet attracted, but might be. My intended audience embraces both types. The overall aim of the book is to answer the question, What is mathematics for? and with its inevitable answer, To delight the mind and help us understand the world.
目录
Preface
Part A Brief Lives
The Ancients
A．1 Thales(ca．625-547 8．C．)
Invented geometry and the concepts of theorem and proof；discovered skepticism as a tool of thought
A．2 Pythagoras(ca．580-500 8．C．)
Pythagorean theorem about fight triangles；irrationality of
A．3 Democritus(ca．460-370 8．C．)
Atoms in physics and mathematics；volume of a cone
A．4 Euclid(ca．300 8．C．)
Organized most of the mathematics known at his time；Euclid’S
theorems on perfect numbers and the infinity of primes
A．5 Archimedes(ca．287-212 8．C.)
Determined volumes，areas and tangents，essentiallyby calculus；found volume and surface area of a sphere；centers of gravity；spiral of Archimedes；calculated Appendix：The Tex of Archimedes
A. 6 Apollonius (ca,262-190 B.C.)
Treatise t3i'f carnie sections-Appendix：Apollonius 'General Preface to His Treatise
A．7 Heron(first century A．D．)
Heron’s principle；area, of a triangle in terms of sides
A.8 Pappus (fourth century A.D)
Centers of gravity linked to solids and surfaces of revolutionAppendix : the focus- Directrix-eccentricity definitions of the conic sections
A.9 Hypatia (A.D. 370?-415)
The first woman mathematicianAppendix: A Proof of Diophantus ' Theorem on PythagoreanTriplesThe Forerunners
A.10 Kepler (1571-1630)
Founded dynamical astronomy; started chain of ideas leadingto integral calculus
A.11 Descartes (1596-1650)
Putative discoverer of analytic geometry; introduced some good notations; first modern philosopher
A.12 Mersenne (1588-1648)
Lubricated the flow of ideas; cycloids; Mersenne primes
A.13 Fermat (1601-1665)
Actual discoverer of analytic geometry; calculated and usedderivatives and integrals; founded modern number theory;probability
A.14 Cavalieri (1598-1647)
Developed Kepler's ideas into an early form of integration
A.15 Torricelli (1608-1647)
Area of cycloid; many calculus problems, even improper
integrals; invented barometer; Torricelli's law in fluid dynamics
A.16 Pascal (1623-1662)
Mathematical induction; binomial coefficients; cycloid; Pascal's
theorem in geometry; probability; influenced Leibniz
A.17 Huygens (1629-1695)
Catenary; cycloid; circular motion; Leibniz's mathematics
teacher (what a pupil! what a teacher!)
The Early Moderns
A.18 Newton (1642-1727)
Invented his own version of calculus; discovered FundamentalTheorem; used infinite series; virtually created physics andastronomy as mathematical sciences
Appendix: Newton's 1714(?) Memorandum of the Two PlagueYears of 1665 and 1666
A. 19 Leibniz (1646-1716)Invented his own better version of calculus; discovered
Fundamental Theorem; invented many good notations; teacherof the Bernoulli brothers
A.20 The Bernoulli Brothers (James 1654-1705, John 1667-1748)
Learned calculus from Leibniz, and developed and applied itextensively; infinite series; John was teacher of Euler
A.21 Euler (1707-1783)
……
Part B Memorable Mathematics
Answers to Problems
Index
