TILE DIFFERENTIAL AND INTEGRAL CALCULUS. This is one of the most useful branches of mathe matics. While elementary algebra and geometry deal with quantities whose value is fixed. the cal culus investigates quantities whose value is con tinually changing. Considering that all nature in all its aspects varies continually, the impor tance of a mathematical method of dealing with variables is evident; and it is easy to see why science had made so little progress before the invention of the calculus, and why progress has been so rapid since.
Three simple examples may serve to show the kind of problems usually attacked by the calcu lus, and the manner in which it solves them. The first two of these examples can also, on account of their simplicity, be solved by means of ele mentary algebra, without resorting to the calcu lus. Nevertheless, they are typical calculus prob lems. and furnish as good examples of the calcu lus method as would be furnished by similar but Dutch more complicated problems lying really beyond the power of elementary mathematics.
Problem I. Suppose the sum of two adjoining sides of a rectangle known. What must be the length of each side so that the rectangle may have the greatest possible area? Problem II. A person in a boat 3 miles from the nearest point on a straight shore wishes to reach a place 5 miles away from that point. Ile can row 4 miles an hour and walk 5 miles an hour. Where should he land in order to reach his point in minimum time? Problem III. To determine the work per formed when a gas is compressed at constant temperature is one of the fundamental problems of theoretical engineering. Work is generally as the force required to move a body, multiplied by the distance traversed. In the case of a gas compressed in a cylindrical vessel, the body moved is the piston. If at the beginning of the experiment the pressure exercised on the pis ton is, say, p pounds per square inch of surface, and the area of the piston is a; then p X a is evi dently the force acting on the piston. This force, however. multiplied by the distance traversed by the piston during compression will not by any means give the work performed. For during compression the force will, of course, have to be continually increased; in other words, it will not retain its original value alp fixed. but will be a variable. In this ease algebra and geometry fail to give a method of direct computation and the calculus has to be resorted to.
In order to understand how the calculus deals with problems of this nature, it is necessary to grasp clearly some fundamental ideas, which usually appear somewhat difficult to the beginner iu calculus, just as the idea of any fixed num ber being represented by the letters a, b, c, ap pears difficult to the child first taking up the study of elementary algebra.
Fundamental Ideas: Function, Differential, Differential Coefficient, Limit.—Va Hal) I es are represented in calculus by the Latin letters x, etc., or by the Greek letters E. r. etc., just as unknown quantities are represented in algebra. If the value of one variable continually depends on that of another variable, the first variable is said to be a function of the second, and the fact is denoted by writing: y = f (x). Thus, the variable area y of a square is said to be a func tion of the variable length or of its side, and in this case the expression y = f (x) stands for the equation y = In investigating the functions and their variables, the calculus catches them at a given moment for the purpose of determining the relative rate of their variation at that 'mo ment. Consider the motion of a ball thrown up
in the air. Its velocity changes from instant to instant. We might get a rough idea of its mo tion by measuring the distance traversed during the first seeond, during. the second second. (hiring the third second, etc. But our results would be far from precise; for, however small an interval of time a second is, the velocity of our ball, changing continually, must be different at the end of that interval from what it is at its be ginning. Our results would be even rougher if instead of the second we employed as a unit of time the minute. To render the results mathe matically precise, we would have to take for our unit not a finite. but an infinitely, small interval of time, an instant. The distance traversed dur ing such an interval would be culled the differ ential of distance and would be denoted in calcu lus by the symbol di. if I stand for distance. Similarly, our infinitely small interval of time would be called the differentia/ of time and would be denoted by the symbol dt, if t stand for time. But as this idea of what a differential is is somewhat vague. owing to the diffirulty of actually conceiving something that is 'infinitely small; the following considerations may be re sorted to. Studying the motion of a ball thrown up in the air, we consider infinitely small inter vals of time cat merely in order to be able to think of the motion as uniform: for within any finite interval the motion is variable. But if at a given instant the motion should actually be come uniform, and continue so, we might think of our differential tit as representing any finite length of time, be it 5 minutes, or 10 minutes, or 500 minutes. For when a boilv moves with perfectly uniform speed, that speed may be read ily determined by ascertaining the distance trav ersed during any interval of time whatever; the result is the same whether we divide the dis traversed in .1 minutes by 5, or that traversed in 10 minutes by 10. We may, accord ingly, define the differential of distance dl as the distanee that 'would br traversed by the ball in an arbitrary. finite interval of time. dl, beginning at a given instant, if at that instant the motion, became uniform. In this manner we may avoid thinking of infinitely small quantities. The velocity would then be dl+dt, no matter now great or small dt is supposed to be. The ratio i — is called the differential eoefficieat of / with t respect to t—the distance 1 being of course 'a function' of the time t. This ratio represents a limit. For, considering again the ball thrown up in the air, the error introduced by choosing a finite inslead of an infinitesimal interval of time Is the less the smaller an interval is chosen, and d finally the true velocity is approached as a limit, when the interval of time becomes infinite ly small. All this is concisely represented by a few symbols, as follows: di 41,0 = dt In this expression AI stands for some finite interval (`increment') of time, and Al for the distance actually traversed during that interval. And the expression tells that when At ap proaches zero (At = 0). i.e. when it becomes in finitely small, the ratio approaches as a limit di • the value It need hardly be remarked that Wit.