INFINITESIMAL CALCULUS 13 PART I. DIFFERENTIAL CALCULUS. 1. In the application of algebra to the theory of curves and sur faces some of the quantities under consideration are conceived as having always the same magnitude, such as the radius of a given circle or of a given sphere, or the axes of a given ellipse or hyperbola ; others again are indefinite, and may have any number of particular values, such as the coordinates of any point on a curve. This difference naturally suggests the division of the quantities involved in any question into two kinds, one called constants, the other variables. It is usual in analysis to denote constants by the first letters of the alphabet, a, b, c, &c. ; variables by the last, u, v, w, x, y, z, &c. 2. One quantity is said to be a function of another when they are so related that any change made in the one causes a corresponding variation in the other. This relation may subsist whether there exist an expression for the function by which its value is determined for each value of its argument ; or the relation may sometimes be defined by certain characteristics of continuity and discontinuity. When an expression is presupposed the relation is usually repre sented by the letters F, /, </>, &c. Thus the equations denote that u, v, w are regarded as functions of x, whose values are determined for any particular value of x when the forms of the functions are known. In each of these expressions the argument x is regarded as the independent variable, to which any value may be assigned at pleasure ; and v, v, w are called dependent variables, as their values depend on that of x, and are determined when it is known. For example, in each of the equations the value of y is known when that of x is given. Such functions are called explicit. 3. In many cases a variable y, instead of being given explicitly in terms of x, is connected with it by an equation of a more com plicated character. For instance, suppose them connected by the relations x log y = y log x, sin y = x sin (a + y}, y 3 + x 3 + 3axy = ; in these cases the value or values of y may be supposed known when x is given, and y is said to be an implicit function of x. Such cases are comprehended in the form In such a form y may be regarded as an implicit function of y, or x as an implicit function of y, at pleasure. 4. Again a quantity may be a function of two or more independent variables. Thus in the equation 7t = sin (ax + by), x and y may be regarded as independent variables, and u as a function of them. Such functions are in general denoted by <t>(x, y <(>(x, y, z), &c. 5. A function (f>(x) is said to bo continuous between any limiting values of x, such as a and b, when to each value of x between those limits there corresponds a finite value of the function, and when an indefinitely small change in the value of x produces only an indefinitely small change in the function. In such cases the func tion in its passage from any one value to any other between the limits receives every intermediate value, and does not become in finite. This continuity can be readily illustrated by taking $(x) as the ordinate of a curve, whose equation may then be written y = <p(x). 6. If the variable x be supposed to receive any change, such change is called an increment ; this increment of x is usually repre sented by the notation Ax. A decrement is regarded as a negative increment. When the increment, or difference, is supposed to be indefinitely small, it is called a differential, and is represented by dx ; i.e., an infinitely small difference is called a differential. In like manner if u be a function of x, and x become x + Ax, the corresponding value of u is denoted by u + Au ; i.e., the increment of u is represented by Au. For finite increments of x it is obvious that the ratio of the increment of u to the corresponding increment of a; has, in general, a finite value. Also when the increment of a; is regarded as being indefinitely small we find that the above men tioned ratio, i.e.,- , has in general in each case a definite limiting dx value ; and the first study of the differential calculus necessarily in volves the investigation of such limiting ratios for the different forms of functions of x. In fact we have seen that the differential calculus took its rise from the investigation of the limiting value of the ratio of the increment of the ordinate y to that of the abscissa x, so as to find tho position of the tangent at any point on a curve. Thus if the equation of a curve, referred to rectangular axes, be denoted by/(x,?/) = 0, then , i.e., the limiting vnlue of ^ for any Ax point on the curve, represents the tangent of the angle which the tangent at the point makes with the axis of x. 7. Again, if we suppose x to become x + h (where h represents Ax, the increment of a;) in the equation u=f(x), then the increment of () hencc is represented by f(x + h) -f(x), and ^1 4 Ax h represents the limit to which da
dx f(x + h)-f(x) h approaches indefinitely, when h is diminished without limit. There are two methods in general of finding this value of The first consists in determining the limiting value o f/( x + ") ~/I- ) h by decreasing h indefinitely. The second consists in expanding f(x + h) in a scries of ascending powers of h, and taking the coeffi cient of h in the expansion. This is the method introduced by Lagrange when he proposed to make the calculus a branch of ordinary algebra, and altogether independent of the consideration of infinitely small magnitudes, or of limits. It is easily seen, as was shown by Lagrange, that the result obtained by the latter method is the same as that arrived at by the former; for, since f(x + h) becomes f(x) when k = 0, f(x) is the first term in the expansion, and we may assume f(x + 70 =f(x) +ph + qh" + &c. , in which p, q, &c. , represent functions of x, independent of /;, then If now we suppose h = 0, the left hand side reduces top ; and, accord ingly the coefficient of h in the expansion off(x + h) is the limiting value of the expression ^ a! + ^ "-^ . This coefficient of h was called by Lagrange the first derived func tion of the original f unction f[x}, and he represented it by the nota tion f(x).
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dx dx Hence we have In this case f (x)dx is called the differential of f(x), and f (x) is called its differential coefficient. 8. We have already seen that the principles of the calculus may be regarded either from the consideration of limits, or from that of infinitesimals or differentials ; the former was the method adopted by Newton, in his later investigations at least ; the latter was that adopted by Leibnitz. The limit of a variable magnitude may be defined as follows. If a variable magnitude tends continually to equality with a certain fixed magnitude, and approaches nearer to it than any assignable difference, however small, this fixed magnitude is called the limit of tlw variable magnitude. For example, if we suppose a polygon inscribed in or circumscribed to any closed curve, and afterwards imagine each side indefinitely diminished, then the closed curve is said to be the limit of either polygon. By this means the whole length of the curve is the limit of the perimeter of either polygon, and the area of the curve is the limit to the area of either polygon. 9. The following principles concerning limits are of frequent application. (1) The limit of the product of two quantities, which vary together, is the product of their limits. (2) The limit of the quotient of two quantities is the quotient of their limits. These are nearly self-evident propositions; they may, however, be formally proved as follows. Let P, Q represent the variable quantities, and p, q their limits ; then, if P=p + a, and Q, = q + &, a, /3 denote quantities which diminish indefinitely as P and Q approach their limits, and become evanescent in the limit. Again, Accordingly in the limit, PQ =pq. The corresponding theorem for the quotients is established easily in like manner. 10. Again, if we conceive any finite number or magnitude to be divided into a very great number of equal parts, each part is very small in comparison with the original magnitude. By supposing the number of parts to be increased indefinitely, i.e., so as to exceejl any assigned number, however great, then each part may be regarded as indefinitely small in comparison with the proposed magnitude, and may be called an infinitesimal with regard to it. By an infinitesimal, or an indefinitely small magnitude, we under stand a magnitude which is less than any assigned magnitude
