数学专业英语－Historical introduction of Calculus

yunjiang · 发表于 2004-5-6 09:40:03

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>The Two Basic Concepts of Calculus
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>The remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in physics,engineering,chemistry,geology,biology, and other fields including,rather recently,some of the social sciences.
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>To give the reader an idea of the many different types of problems that can be treatedby the methods of calculus,we list here a few sample questions.
With what speed should a rocket be fired upward so that it never returns to earth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter Ｌ? What volume of material is removed from a solid sphere of radius 2 r if a hole of redius r is drilled through the center? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at the end of two hours? If a ten-pound force stretches an elastic spring one inch,how much work is required to stretch the spring one foot?
These examples,chosen from various fields,illustrate some of the technical questions that can be answered by more or less routine applications of calculus.
Calculus is more than a technical tool－it is a collection of fascinating and exeiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcepts from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is its unifying power.Most of these ideas can be formulated so that they revolve around two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.
Consider a cruve C which lies above a horizontal base line such as that shown in Fig.1. We assume this curve has the property that every vertical line intersects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two parallel vertical segments joining C to the base.The first fundamental problem of calculus is this: To assign a number which measures the area of this shaded region.
 Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which measures the steepness of this line.
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Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent line and to calculate the area of a given region or the steepness of a given angent line. Integral calculus deals with the problem of area while differential calculus deals with the problem of tangents.
 
Historical Background
The birth of integral calculus occurred more than 2000 years ago when the Greeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be described briefly as follows:Given a region whose area is to be determined,we inscribe in it a polygonal region which approximates the given region and whose area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Archimedes(287－212 B.C.) to find exact formulas for the area of a circle and a few other special figures.
 The development of the method of exhaustion beyond the point to which Archimcdcs carried it had to wait nearly eighteen centuries until the use of algebraic symbols and techniques became a standard part of mathematics. The elementary algebra that is familiar to most high-school students today was completely unknown in Archimedes’ time,and it would have been next to impossible to extend his method to any general class of regions without some convenient way of expressing rather lengthy calculations in a compact and simpolified form.
A slow but revolutionary change in the development of mathematical notations began in the 16 th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbols “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant successe of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding algebraic solutions of cubic and quadratic equations stimulated a great deal of activity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion and a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis.
 
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Fig.2. The method of exhaustion applied to a semicircular region.
 
Gradually the method of exhaustion was transformed into the subject now called integral calculus,a new and powerful discipline with a large variety of applications, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retained some of the original features of the method of exhaustion,received its biggest impetus in the 17 th century, largely due to the efforts of Isaac Newion (1642—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riemann (1826-1866).Further refinements and extensions of the theory are still being carried out in contemporary mathematics.

yunjiang · 发表于 2004-5-6 09:40:31

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