Page:EB1911 - Volume 14.djvu/568

are either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients.

Again let F(x) be the indefinite integral of a continuous function ƒ(x), so that we have

${\frac {dF(x)}{dx}}=f(x),\int _{a}^{b}f(x)\,dx=F(b)-F(a)$

When the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x₁, x₂, ... x_n−1 be the abscissae of the points in order. The integral is the limit of the sum

$f(a)(x_{1}-a)+f(x_{1})(x_{2}-x_{1})+\ldots +f(x_{r})(x_{r+1}-x_{r})+\ldots +f(x_{n-1})(b-x_{n-1})$

every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences

${F(x_{1})-F(a)}+{F(x_{2})-F(x_{1})}+\ldots +{F(x_{r+1})-F(x_{r})}+\ldots +{F(b)-F(x_{n-1})}$

for this sum is F(b) − F(a). Now the difference F(x_r+1) − F(x_r) is not equal to the differential ƒ(x_r) (x_r+1 − x_r), but the sum of the differences is equal to the limit of the sum of these differentials. The differential may be regarded as so much of the difference as is required to form the integral. From this point of view a differential is called a differential element of an integral, and the integral is the limit of the sum of differential elements. In like manner the differential element ydx of the area of a curve (§ 5) is not the area of the portion contained between two ordinates, however near together, but is so much of this area as need be retained for the purpose of finding the area of the curve by the limiting process described.

8. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letter “d ” is the initial letter of the word differentia (difference) and the symbol “∫” is a conventionally written “S”, the initial letter of the word summa Notation.(sum or whole). The notation was introduced by Leibnitz (see §§ 25-27, below).

9. The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x+Δx for x and y+Δy for y. If there is a differential coefficient Fundamental Artifice.of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim._{Δx = 0} (R/Δx) = 0, as in § 7 above. The artifice consists in rejecting ab initio all terms of the equation which belong to R. We do not form R at all, but only φ.Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y.dx is formed. In fig. 3 of § 5 the element of area y.dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we have

${\begin{matrix}\lim _{MN\to 0}{\frac {MN.QR}{MN}}\end{matrix}}=0.$

Thus the artifice by which differential elements of integrals are formed is in principle the same as that by which differentials are formed without first forming differential coefficients.

10. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which are connected together by any relation, and if when x tends to zero y also tends to zero, the fraction y/x may tend to a finite limit. In this case x and y are said to be “of the Orders of small quantities.same order.” When this is not the case we may have either

${\begin{matrix}\lim _{x\to 0}{\frac {x}{y}}\end{matrix}}=0,$

or

${\begin{matrix}\lim _{x\to 0}{\frac {y}{x}}\end{matrix}}=0.$

In the former case y is said to be “of a lower order” than x; in the latter case y is said to be “of a higher order” than x. In accordance with this notion we may say that the fundamental artifice of the infinitesimal calculus consists in the rejection of small quantities of an unnecessarily high order. This artifice is now merely an incident in the conduct of a limiting process, but in the 17th century, when limiting processes other than the Greek methods for quadratures were new, the introduction of the artifice was a great advance.

11. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain the rules by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as Rules of Differentiation. follows:—

(i.) The differential coefficient of a constant is zero
(ii.) For a sum u+v+ . . ., where u,v,... are functions of x,

${\frac {d(u+v+\ldots +z)}{dx}}={\frac {du}{dx}}+{\frac {dv}{dx}}+\ldots +{\frac {dz}{dx}}$

(iii.) For a product uv

${\frac {d(uv)}{dx}}=u{\frac {dv}{dx}}+v{\frac {du}{dx}}.$

(iv.) For a quotient u/v

${\frac {d(u/v)}{dx}}=\left(v{\frac {du}{dx}}-u{\frac {dv}{dx}}\right){\Bigg /}v^{2}.$

(v.) For a function of a function, that is to say, for a function y expressed in terms of a variable z, which is itself expressed as a function of x,

${\frac {dy}{dx}}={\frac {dy}{dz}}\cdot {\frac {dz}{dx}}.$

In addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table:—

$y\,$	${\frac {dy}{dx}}$
$x^{n}\,$	${\begin{matrix}nx^{n-1}\\{\mbox{ for all values of }}n\end{matrix}}$
$\log _{a}x\,$	$x^{-1}log_{a}e\,$
$a^{x}$	$a^{x}log_{e}a\,$
$\sin {x}\,$	$\cos {x}\,$
$\cos {x}\,$	$-\sin {x}\,$
$\sin ^{-1}x\,$	$(1-x^{2})^{-{\tfrac {1}{2}}}\,$
$\tan ^{-1}x\,$	$(1+x^{2})^{-1}\,$

Each of the formal rules, and each of the particular results in the table, is a theorem of the differential calculus. All functions (or rather expressions) which can be made up from those in the table by a finite number of operations of addition, subtraction, multiplication or division can be differentiated by the formal rules. All such functions are called explicit functions. In addition to these we have implicit functions, or such as are determined by an equation containing two variables when the equation cannot be solved so as to exhibit the one variable expressed in terms of the other. We have also functions of several variables. Further, since the derived function of a given function is itself a function, we may seek to differentiate it, and thus there arise the second and higher differential coefficients. We postpone for the present the problems of differential calculus which arise from these considerations. Again, we may have explicit functions which are expressed as the results of limiting operations, or by the limits of the results obtained by performing an infinite number of algebraic operations upon the simple functions. For the problem of differentiating such functions reference may be made to Function.

12. The processes of the integral calculus consist largely in transformations of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we Indefinite Integrals. have those in the following table:—

$f(x)\,$	$\int f(x)dx$
$x^{n}\,$	${\begin{matrix}{\frac {x^{n+1}}{n+1}}\\{\mbox{ for all values of }}n{\mbox{ except }}-1\end{matrix}}$
${\frac {1}{x}}\,$	$\log _{e}x\,$
$e^{ax}\,$	$a^{-1}e^{ax}\,$
$\cos {x}\,$	$\sin {x}\,$
$\sin {x}\,$	$-\cos {x}\,$
$(a^{2}-x^{2})^{-{\frac {1}{2}}}\,$	$\sin ^{-1}{\frac {x}{a}}\,$
${\frac {1}{a^{2}+x^{2}}}$	${\frac {1}{a}}\tan ^{-1}{\frac {x}{a}}$

The formal rules of § 11 give us means for the transformation of integrals into recognizable forms. For example, the rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms. The rule (iii.) for a product leads to the method of integration by parts. The rule (v.) for a function of a function leads to the method of substitution (see § 48 below).