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[OUTLINES
INFINITESIMAL CALCULUS


finding out whether this condition is satisfied or not is to attempt to form the differential coefficient. If the quotient of differences Ay/Ax hasa limit when Ax tends to zero, yis a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function f(x) is always defined by the formula

dx ° h

Rules for the formation of differential coefficients in particular cases have been given in § II 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 has one value when h approaches zero through positive values, and a different value when h approaches zero coefficient

52132; through negative values. The two limits are then called Regresswe the "progressive 'I and ' regressive " differential co-Dufemm efiie1ent s. In applications to dynamics, when x denotes Ha, Co a coordinate and t the time, dx/dt denotes a velocity. If emdents 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 Ar“s 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 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 fydx 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 Rl, R2, ...R, , 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 PR;-I-R1R2+ . . . -I-R, , 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 inde-Lengths finitely, this limit is the length of the are PQ. The limit “curves is th e same whatever law may be adopted for inserting the intermediate points R and diminishing the lengths of the chords It appears from this statement that the differential element cf 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, IO), the differential element of arc ds may be expressed by the formula

dS = / {(dx)2+(<1y)2l,

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 f'(x) vanishes at all points of an interval, the function f(x) is constant in the interval. It follows E?;"f:;"ts that, if two functions have the same derived function they can only differ by a constant. Conversely, indefinite grafion. . . . .

integrals are indeterminate to the extent of an additive Constant.

35. The differential coefficient dy/dx, or the derived function f'(x), is itself a function of x, and its differential coefficient is denoted HI her by f'f(x) or d'y/dxz. In the second of these notations Dlgerem dfdx IS regarded as the symbol of an operation, that of “Mem differentiation with respect to x, and the Index 2 means emdmts that the operation is repeated. In like manner we may express the results of n successive differentiations by f< ) (rc) or by d"y/dx". When the second differential coefficient exists, or the first is differentiable, we have the relation f”<x> =1;m., .../1%-fl-1” “' 22” “L -f°° ) <i.> The limit expressed by the right-hand member of this equation may exist in cases in which f'(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, thenf(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<"7(x) =lim-1t.=oh"" f(x-|-nh) - nf{(x-{-(11.- I)h} + f{x+<n-2>1.; - + (i>»f(x>]» (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 f'(x), f”(x), . . .f<"">(x) do not exist. Corresponding to the rule iii. of § II we have the rule for forming the nth differential coefficient of a product in the form d"(uv) d"v du d"“11J n(n-1) ilu d""2v d"u dx” “dx»+”dx dx~~L+ 1.2 dxf dx"-2+ ' ' 'JV dxf where the coefficients are those of the expansion of (I -I-x)" in powegs 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=f(x), the nth differential d" y is defined by the equation dw =f<"><x><¢x>»,

in which dx is the (arbitrary) differential of x. /Vhen d/dx is regarded as a single symbol of operation the symbol f...dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D'1. D" means that S b I the operation D is to be performed n times in succession; of? gi D“" that the operation of forming the indefinite integral “tmp 3 is to be performed n times in succession. Leibnitz's course of thought (§ 24) naturally led him to inquire after an interpretation of D". 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.

36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of intermediate value (“ theorem of mean

Theorem B

value, ':Y"Utheorem of finite incre- “flute” P ments, Rolle s theorem, are mediate other names for it . This theorem

Value.

may be explained as follows: A

v=f(x)

Let A, B be two points of a curve

(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 f'(x) is and b, there is a value xl of x between a and b which has the property expressed by the equation

FIG. 9.

continuous between a

f 2-f'<x.>. <i.>

The value xl can be expressed in the form a-I-0(1)-a) where 0 is a number between o and 1.

A slightly more general theorem was given by Cauchy (1823) to the effect that, if f'(x) and F'(x) are continuous between x=a and x = b, then there is a numbcr6 between 0 and I which has the property expressed by the equation

Ffh) ~ F(fl) = F'lll-I-0(b ~ all

f(11) -fffl) f'lf1+9(b *Ulf

The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where f(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 Fw; Fd) = bF'(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 f(x) and all its differential coefficients up to the nth inclusive are continuous in the interval betweenx=a and!-Qb, then there is a number 0 between o and I which has the property expressed by the equation

f<1»> -f<<1>+<b - ¢1)f'(¢1)+%'{3Xf”(f1) +. . +$£§ f<~-1><a> +g%if""{<1+0(b-<1)}- <i.>

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 T I, point are known. The function is expressed by a termin~ Tzy or S ated series, and, when the remainder tends to zero as n eorem

increases, it may be transformed into an infinite series. The theorem