1911 Encyclopædia Britannica/Infinitesimal Calculus/Outlines

II. History (§23-31)

Infinitesimal Calculus III. Outlines of the Infinitesimal Calculus (§32-39)

III. Outlines of the Infinitesimal Calculus (§40-46)

III. Outlines of the Infinitesimal Calculus (§32-39)

III. Outlines of the Infinitesimal Calculus.

32. The general notions of functionality, limits and continuity are explained in the article Function. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.

33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x = t. If certain conditions are satisfied the aggregate of the points determined by the functional relations form a curve. The Geometrical limits. first condition is that the aggregate of the values of t to which values of x and y correspond must be continuous, or, in other words, that these values must consist of all real numbers, or of all those real numbers which lie between assigned extreme numbers. When this condition is satisfied the points are “ordered,” and their order is determined by the order of the numbers t, supposed to be arranged in order of increasing or decreasing magnitude; also
Fig. 8. there are two senses of description of the curve, according as t is taken to increase or to diminish. The second condition is that the aggregate of the points which are determined by the functional relations must be “continuous.” This condition means that, if any point P determined by a value of t is taken, and any distance δ, however small, is chosen, it is possible to find two points Q, Q′ of the aggregate which are such that (i.) P is between Q and Q′, (ii.) if R, R′ are any points between Q and Q′ the distance RR′ is less than δ. The meaning of the word “between” in this statement is fixed by the ordering of the points. Sometimes additional conditions are imposed upon the functional relations before they are regarded as defining a curve. An aggregate of points which satisfies the two conditions stated above is sometimes called a “Jordan curve.” It by no means follows that every curve of this kind has a tangent. In order that the curve may have a tangent at P it is necessaryTangents. that, if any angle α, however small, is specified, a distance δ can be found such that when P is between Q and Q′, and PQ and PQ′ are less than δ, the angle RPR′ is less than α for all pairs of points R, R′ which are between P and Q, or between P and Q′ (fig. 8). When this condition is satisfied y is a function of x which has a differential coefficient. The only way of finding out whether this condition is satisfied or not is to attempt to form the differential coefficient. If the quotient of differences Δy/Δx has a limit when Δx tends to zero, y is a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function ƒ(x) is always defined by the formula

$f^{\prime }(x)={\frac {df(x)}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$.

Rules for the formation of differential coefficients in particular cases have been given in §11 above. The definition of a differential coefficient, and the rules of differentiation are quite independent of any geometrical interpretation, such as that concerning tangents to a curve, and the tangent to a curve is properly defined by means of the differential coefficient of a function, not the differential coefficient by means of the tangent.

It may happen that the limit employed in defining the differential coefficient has one value when h approaches zero through positive values, and a different value when h approaches zero through negative values. The two limits are then called the “progressive” and “regressive” differential coefficients. Progressive and Regressive Differ-ential Coefficients. In applications to dynamics, when x denotes a coordinate and t the time, dx/dt denotes a velocity. If the velocity is changed suddenly the progressive differential coefficient measures the velocity just after the change, and the regressive differential coefficient measures the velocity just before the change. Variable velocities are properly defined by means of differential coefficients.

All geometrical limits may be specified in terms similar to those employed in specifying the tangent to a curve; in difficult cases they must be so specified. Geometrical intuition may fail to answer the question of the existence or non-existence of the appropriate limits. In the last resort the definitions of many Areas. quantities of geometrical import must be analytical, not geometrical. As illustrations of this statement we may take the definitions of the areas and lengths of curves. We may not assume that every curve has an area or a length. To find out whether a curve has an area or not, we must ascertain whether the limit expressed by ∫ydx exists. When the limit exists the curve has an area. The definition of the integral is quite independent of any geometrical interpretation. The length of a curve again is defined by means of a limiting process. Let P, Q be two points of a curve, and R₁, R₂, . . . R_n−1 a set of intermediate points of the curve, supposed to be described in the sense in which Q comes after P. The points R are supposed to be reached successively in the order of the suffixes when the curve is described in this sense. We form a sum of lengths of chords

PR₁ + R₁R₂ + . . . + R_n−1Q.

If this sum has a limit when the number of the points R is increased indefinitely and the lengths of all the chords are diminished indefinitely, this limit is the length of the arc PQ. The limit is the same whatever law may be adopted for inserting the intermediate points R and diminishing the lengthsLengths of Curves. of the chords. It appears from this statement that the differential element of the arc of a curve is the length of the chord joining two neighbouring points. In accordance with the fundamental artifice for forming differentials (§§ 9, 10), the differential element of arc ds may be expressed by the formula

$ds={\sqrt {(dx)^{2}+(dy)^{2}}}\,$,

of which the right-hand member is really the measure of the distance between two neighbouring points on the tangent. The square root must be taken to be positive. We may describe this differential element as being so much of the actual arc between two neighbouring points as need be retained for the purpose of forming the integral expression for an arc. This is a description, not a definition, because the length of the short arc itself is only definable by means of the integral expression. Similar considerations to those used in defining the areas of plane figures and the lengths of plane curves are applicable to the formation of expressions for differential elements of volume or of the areas of curved surfaces.

34. In regard to differential coefficients it is an important theorem that, if the derived function ƒ′(x) vanishes at all points of an interval, the function ƒ(x) is constant in the interval. It follows that, if two functions have the same derived function they can only differ by a constant. Conversely, indefinite integrals are indeterminate to the Constants of Integration. extent of an additive constant.

35. The differential coefficient dy/dx, or the derived function ƒ′(x), is itself a function of x, and its differential coefficient is denoted by ƒ″(x) or d²y/dx². In the second of these notations d/dx is regarded as the symbol of an operation, that of differentiation with respect to x, and the index 2 means Higher Differential Coefficients. that the operation is repeated. In like manner we may express the results of n successive differentiations by ƒ⁽ⁿ⁾(x) or by dⁿy/dxⁿ. When the second differential coefficient exists, or the first is differentiable, we have the relation

$f^{\prime \prime }(x)=lim_{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}$

(i.)

The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′(x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ(x) must be a linear function of x in the interval, is important.

The relation (i.) is a particular case of the more general relation

$f^{(n)}(x)=\lim _{h\to 0}h^{-n}{\bigg [}f(x+nh)-nf\left\{x+(n-1)h\right\}+{\frac {n(n-1)}{2!}}f\left\{x+(n-2)h\right\}-\ldots +(-1)^{n}f(x){\bigg ]}$

(ii.)

As in the case of relation (i.) the limit expressed by the right-hand member may exist although some or all of the derived functions ƒ′(x), ƒ″(x), . . . ƒ⁽ⁿ⁻¹⁾(x) do not exist.

Corresponding to the rule iii. of § 11 we have the rule for forming the nth differential coefficient of a product in the form

${\frac {d^{n}(uv)}{dx^{n}}}={\frac {d^{n}v}{dx^{n}}}+n{\frac {du}{dx}}{\frac {d^{n-1}v}{dx^{n-1}}}+{\frac {n(n-1)}{1\cdot 2}}{\frac {d^{2}u}{dx^{2}}}{\frac {d^{n-2}v}{dx^{n-2}}}+\ldots +{\frac {d^{n}u}{dx^{n}}}v$,

where the coefficients are those of the expansion of (1 + x)ⁿ in powers of x (n being a positive integer). The rule is due to Leibnitz, (1695).

Differentials of higher orders may be introduced in the same way as the differential of the first order. In general when y = ƒ(x), the nth differential dⁿy is defined by the equation

dⁿy = ƒ⁽ⁿ⁾(x)(dx)ⁿ,

in which dx is the (arbitrary) differential of x.

When d/dx is regarded as a single symbol of operation the symbol ∫. . .dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D⁻¹. Dⁿ means that the operation D is to be performed n times in succession; D⁻ⁿ that the operation of forming the indefinite integralSymbols of operation. is to be performed n times in succession. Leibnitz’s course of thought (§ 24) naturally led him to inquire after an interpretation of Dⁿ where n is not an integer. For an account of the researches to which this inquiry gave rise, reference may be made to the article by A. Voss in Ency. d. math. Wiss. Bd. ii. A, 2 (Leipzig, 1889). The matter is referred to as “fractional” or “generalized” differentiation.

Fig. 9.

36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of intermediate value (“theorem of mean value,” “theorem of finite increments,” “Rolle’s theorem,” are other names for it). This theorem may be explained as follows: Theorem of Intermediate Value. Let A, B be two points of a curve y = ƒ(x) (fig. 9). Then there is a point P between A and B at which the tangent is parallel to the secant AB. This theorem is expressed analytically in the statement that if ƒ′(x) is continuous between a and b, there is a value x₁ of x between a and b which has the property expressed by the equation

${\frac {f(b)-f(a)}{b-a}}=f^{\prime }(x_{1})$.

(i.)

The value x₁ can be expressed in the form a + θ(b − a) where θ is a number between 0 and 1.

A slightly more general theorem was given by Cauchy (1823) to the effect that, if ƒ′(x) and F′(x) are continuous between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

${\frac {F(b)-F(a)}{f(a)-f(b)}}={\frac {F^{\prime }\left\{a+\theta (b-a)\right\}}{f^{\prime }\left\{a+\theta (b-a)\right\}}}$.

The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where ƒ(x) is a rational integral function which vanishes when x = a and also when x = b. The general theorem was given by Lagrange (1797). Its fundamental importance was first recognized by Cauchy (1823). It may be observed here that the theorem of integral calculus expressed by the equation

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

follows at once from the definition of an integral and the theorem of intermediate value.

The theorem of intermediate value may be generalized in the statement that, if ƒ(x) and all its differential coefficients up to the nth inclusive are continuous in the interval between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

$f(b)=f(a)+(b-a)f^{\prime }(a)+{\frac {(b-a)^{2}}{2!}}f^{\prime \prime }(a)+\ldots$

$+{\frac {(b-a)^{n-1}}{(n-1)!}}f^{(n-1)}(a)+{\frac {(b-a)^{n}}{n!}}f^{(n)}(a)\left\{a+\theta (b-a)\right\}$.

(ii.)

37. This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated series, and, when the remainder tends to zero as n Taylor’s Theorem. increases, it may be transformed into an infinite series. The theorem was first given by Brook Taylor in his Methodus Incrementorum (1717) as a corollary to a theorem concerning finite differences. Taylor gave the expression for ƒ(x + z) in terms of ƒ(x), ƒ′(x), . . . as an infinite series proceeding by powers of z. His notation was that appropriate to the method of fluxions which he used. This rule for expressing a function as an infinite series is known as Taylor’s theorem. The relation (i.), in which the remainder after n terms is put in evidence, was first obtained by Lagrange (1797). Another form of the remainder was given by Cauchy (1823) viz.,

${\frac {(b-a)^{n}}{(n-1)!}}(1-\theta )^{n-1}f^{n}\left\{a+\theta (b-a)\right\}$.

The conditions of validity of Taylor’s expansion in an infinite series have been investigated very completely by A. Pringsheim (Math. Ann. Bd. xliv., 1894). It is not sufficient that the function and all its differential coefficients should be finite at x = a; there must be a neighbourhood of a within which Cauchy’s form of the remainder tends to zero as n increases (cf. Function).

An example of the necessity of this condition is afforded by the function ƒ(x) which is given by the equation

$f(x)={\frac {1}{1+x^{2}}}+\sum _{n=1}^{n=\infty }{\frac {(-1)^{n}}{n!}}{\frac {1}{1+3^{2n}x^{2}}}$.

(i.)

The sum of the series

$f(0)+xf^{\prime }(0)+{\frac {x^{2}}{2!}}f^{\prime \prime }(0)+\ldots$

(ii.)

is the same as that of the series

$e^{-1}-x^{2}e^{-3^{2}}+x^{4}e^{-3^{4}}-\ldots$

It is easy to prove that this is less than e⁻¹ when x lies between 0 and 1, and also that ƒ(x) is greater than e⁻¹ when x = 1/√3. Hence the sum of the series (i.) is not equal to the sum of the series (ii.).

The particular case of Taylor’s theorem in which a = 0 is often called Maclaurin’s theorem, because it was first explicitly stated by Colin Maclaurin in his Treatise of Fluxions (1742). Maclaurin like Taylor worked exclusively with the fluxional calculus.

Examples of expansions in series had been known for some time. The series for log (1 + x) was obtained by Nicolaus Mercator (1668) by expanding (1 + x)⁻¹ by the method of algebraic division, and integrating the series term by term. He regarded his result as a “quadrature of the hyperbola.” Expansions in
power series. Newton (1669) obtained the expansion of sin⁻¹x by expanding (1 − x²)^−1/2 by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan⁻¹x. Newton also obtained the series for sin x, cos x, and e^x by reversion of series (1669). The symbol e for the base of the Napierian logarithms was introduced by Euler (1739). All these series can be obtained at once by Taylor’s theorem. James Gregory found also the first few terms of the series for tan x and sec x; the terms of these series may be found successively by Taylor’s theorem, but the numerical coefficient of the general term cannot be obtained in this way.

Taylor’s theorem for the expansion of a function in a power series was the basis of Lagrange’s theory of functions, and it is fundamental also in the theory of analytic functions of a complex variable as developed later by Karl Weierstrass. It has also numerous applications to problems of maxima and minima and to analytical geometry. These matters are treated in the appropriate articles.

The forms of the coefficients in the series for tan x and sec x can be expressed most simply in terms of a set of numbers introduced by James Bernoulli in his treatise on probability entitled Ars Conjectandi (1713). These numbers B₁, B₂, . . . called Bernoulli’s numbers, are the coefficients so denoted in the formula

${\frac {1}{e^{x}-1}}=1-{\frac {x}{2}}+{\frac {{\text{B}}_{1}}{2!}}x^{2}-{\frac {{\text{B}}_{2}}{4!}}x^{4}+{\frac {{\text{B}}_{3}}{6!}}x^{6}-\ldots ,$

and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type

${\text{B}}_{n}={\frac {(2n)!}{2^{2n-1}\pi ^{2n}}}\left({\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+\ldots \right).$

The function

$x^{m}-{\frac {m}{2}}x^{m-1}+{\frac {m\cdot m-1}{2!}}B_{1}x^{m-2}-\ldots$

has been called Bernoulli’s function of the mth order by J. L. Raabe (Crelle’s J. f. Math. Bd. xlii., 1851). Bernoulli’s numbers and functions are of especial importance in the calculus of finite differences (see the article by D. Seliwanoff in Ency. d. math. Wiss. Bd. i., E., 1901).

When x is given in terms of y by means of a power series of the form

$x=y\left({\text{C}}_{0}+{\text{C}}_{1}y+{\text{C}}_{2}y^{2}+\ldots \right)\quad \left({\text{C}}_{0}\neq 0\right)=yf_{0}(y),{\mbox{ say,}}$

there arises the problem of expressing y as a power series in x. This problem is that of reversion of series. It can be shown that provided the absolute value of x is not too great,

$y={\frac {x}{f_{0}(0)}}+\sum _{n=2}^{N=\infty }\left\lbrack {\frac {x^{n}}{n!}}\cdot {\frac {d^{n-1}}{dy^{n-1}}}{\frac {1}{\left\{f_{0}(y)\right\}^{n}}}\right\rbrack _{y=0}$

To this problem is reducible that of expanding y in powers of x when x and y are connected by an equation of the form

$y=a+xf(y),\,$

for which problem Lagrange (1770) obtained the formula

$y=a+xf(a)+\sum _{n=2}^{N=\infty }\left\lbrack {\frac {x^{n}}{n!}}\cdot {\frac {d^{n-1}}{da^{n-1}}}\left\{f(a)\right\}^{n}\right\rbrack .$

For the history of the problem and the generalizations of Lagrange’s result reference may be made to O. Stolz, Grundzüge d. Diff. u. Int. Rechnung, T. 2 (Leipzig, 1896).

Fig. 10.

38. An important application of the theorem of intermediate value and its generalization can be made to the problem of evaluating certain limits. If two functions φ(x) and ψ(x) both vanish at x = a, the fraction φ(x)/ψ(x) may have a finite limit at a. This limit is described as the limit of an Indeterminate
forms. “indeterminate form.” Such indeterminate forms were considered first by de l’Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y = φ(x)/ψ(x) near the ordinate x = a, when the curves y = φ(x) and y = ψ(x) both cross the axis of x at the same point as this ordinate. In fig. 10 PA and QA represent short arcs of the curves φ, ψ, chosen so that P and Q have the same abscissa. The value of the ordinate of the corresponding point R of the compound curve is given by the ratio of the ordinates PM, QM. De l’Hospital treated PM and QM as “infinitesimal,” so that the equations PM:AM =φ’(a) and QM:AM = ψ′(a) could be assumed to hold, and he arrived at the result that the “true value” of φ(a)/ψ(a) is φ′(a)/ψ′(a). It can be proved rigorously that, if ψ′(x) does not vanish at x = a, while φ(a) = 0 and ψ(a) = 0, then

$\lim _{x=a}{\frac {\phi (x)}{\psi (x)}}={\frac {\phi ^{\prime }(x)}{\psi ^{\prime }(x)}}.$

It can be proved further if that φ^m(x) and ψⁿ(x) are the differential coefficients of lowest order of φ(x) and ψ(x) which do not vanish at x = a, and if m = n, then

$\lim _{x=a}{\frac {\phi (x)}{\psi (x)}}={\frac {\phi ^{n}(x)}{\psi ^{n}(x)}}.$

If m>n the limit is zero; but if m<n the function represented by the quotient φ(x)/ψ(x) “becomes infinite” at x = a. If the value of the function at x = a is not assigned by the definition of the function, the function does not exist at x = a, and the meaning of the statement that it “becomes infinite” is that it has no finite limit. The statement does not mean that the function has a value which we call infinity. There is no such value (see Function).

Such indeterminate forms as that described above are said to be of the form 0/0. Other indeterminate forms are presented in the form 0 × ∞, or 1^∞, or ∞/∞, or ∞ − ∞. The most notable of the forms 1^∞ is lim._x=0(1 + x)^1/x, which is e. The case in which φ(x) and ψ(x) both tend to become infinite at x = a is reducible to the case in which both the functions tend to become infinite when x is increased indefinitely. If φ′(x) and ψ′(x) have determinate finite limits when x is increased indefinitely, while φ(x) and ψ(x) are determinately (positively or negatively) infinite, we have the result expressed by the equation

$\lim _{x=\infty }{\frac {\phi (x)}{\psi (x)}}={\frac {\lim _{x=\infty }\phi ^{\prime }(x)}{\lim _{x=\infty }\psi ^{\prime }(x)}}.$

For the meaning of the statement that φ(x) and ψ(x) are determinately infinite reference may be made to the article Function. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. Such a scale is

$\log x,\ldots x,x^{2},\ldots x^{n},\ldots e^{x},\ldots x^{x};$

each of the limits expressed by such forms as lim._x=∞ φ(x)/ψ(x), where φ(x) precedes ψ(x) in the scale, is zero. The construction of such scales, along with the problem of constructing a complete scale was discussed in numerous writings by Paul du Bois-Reymond (see in particular, Math. Ann. Bd. xi., 1877). For the general problem of indeterminate forms reference may be made to the article by A. Pringsheim in Ency. d. math. Wiss. Bd. ii., A. 1 (1899). Forms of the type 0/0 presented themselves to early writers on analytical geometry in connexion with the determination of the tangents at a double point of a curve; forms of the type ∞/∞ presented themselves in like manner in connexion with the determination of asymptotes of curves. The evaluation of limits has innumerable applications in all parts of analysis. Cauchy’s Analyse algébrique (1821) was an epoch-making treatise on limits.

If a function φ(x) becomes infinite at x = a, and another function ψ(x) also becomes infinite at x = a in such a way that φ(x)/ψ(x) has a finite limit C, we say that φ(x) and ψ(x) become “infinite of the same order.” We may write φ(x) = Cψ(x) + φ₁(x), where lim._x=aφ₁(x)/ψ(x) = 0, and thus φ₁(x) is of a lower order than φ(x); it may be finite or infinite at x = a. If it is finite, we describe Cψ(x) as the “infinite part” of φ(x). The resolution of a function which becomes infinite into an infinite part and a finite part can often be effected by taking the infinite part to be infinite of the same order as one of the functions in the scale written above, or in some more comprehensive scale. This resolution is the inverse of the process of evaluating an indeterminate form of the type ∞ − ∞.

For example lim._x=0{(e^x−1)⁻¹−x⁻¹} is finite and equal to = 1/2, and the function (e^x−1)⁻¹−x⁻¹ can be expanded in a power series in x.

39. The nature of a function of two or more variables, and the meaning to be attached to continuity and limits in respect of such functions, have been explained under Function. The theorems of differential calculus which relate to such functions are in general the same whether the number Functions of
several variables. of variables is two or any greater number, and it will generally be convenient to state the theorems for two variables.