1550078481-Ordinary_Differential_Equations__Roberts_

2 Ordinary Differential Equations

which is equivalent to finding tangents to the curves; he also introduces into

mathematics t he custom of using letters which appear first in the alp habet for

constants and letters which appear last for variables; and he formulates our

present system of exponents in which x^2 denotes x · x , in which x^3 denotes

x · x · x, etc. Pierre de Fermat's claim to priority in the invention of a nalytic

geometry is b ased on a letter he wrote to Gilles Roberval in Septemb er 1636,

in which he claims that the ideas he is advancing are seven years old. Fermat's

method for finding a tangent to a curve was devised in conjunction with his

procedure for determining maxima and minima. Thus, Fermat was the first

mathematician to develop the central idea of differential calculus- t h e notion

of a derivative. Fermat also had great success in the t heory of integration. By

1636 or earlier, Fermat had discovered and proved by geometrical means the

power formula for positive integer exponents- that is, Fermat had proved for

positive integers n

xndx= --.

l

a an+l

0 n+l

Later, Fermat generali zed this result to rational exponents n-=/=--1. In many

respects, t he work of Descartes and Fermat were antipodal. Generally speak-

ing, Descartes started with a locus and then derived its equation, whereas

Fermat began with an equation and t hen found the locus. Their combined

efforts illustrate the two fundamental and inverse properties of analytic ge-

ometry.

Until approximately the middle of the seventeenth century, integral and

differential calculus appeared to be two distinct branches of mathematics.

Integration, in the case of calculating the area under a curve, consisted of

finding the limit approached by the sum of a very large number of extremely

thin rectangles as the number of rectangles increased indefinitely and the

widt h of each rectangle approached zero. Different iation, on the other hand,

consisted of finding the limit of a difference quot ient. About 1646, Evangelista

Torricelli (1608-1647), a student of Galileo and inventor of the barometer

(1643), showed that integration and differentiation were inverse operations

for equations of the form y = xn where n is a positive integer. That is,

Torricelli showed for n a posit ive integer

• d 1x tndt--d ( --xn+l)
  dx o dx n+l

Isaac Barrow (1630-1677) is usually given credit for being t he first mathemati-
cia n to recognize in its fullest generality that differentiation and integration
are inverse operations. In his Lectiones, Barrow essentially stated and proved

geometrically the fundamental theorem of calculus- that is, if f(x) is a

continuous function on the interval [a, b] and if x is in the interval [ a , b], then

dx d 1x a f(t) dt = f(x).