f yields a result definite only as to form, up to an additive constant, C. (Cf. fč Evolute and Involute, above). Derivation simplifies, reducing even transcendents to algebraics; Integration complicates, lifting algebraics and even rationals up into tran4. Derivation is deductive and can create no new forms. Integration is inductive and creates an ∞ of new forms, all defined as integrals. scendents (as √a2-x2 and x 2 cos x cos x dx=sin x cos x + =sin x cos x + x· 2 Jsi 2 sinx.dr cos xdx; whence cos x .dx=x+sin x cos x Similarly for any integral power of sine or cosine except (sin x)-1, (cos x)-', which are reduced by passing to the half-angle, . x What is the range of such reductions? What functions can we thus integrate in terms of known functions? Few enough. Of Algebraics, (x) $(x)' I. Rational functions, tion into part-fractions. by decomposi II. Rational functions of x and (ax+1)a/B Operating directly on y=f(x) by a series of differentiations as to x, say (D), we get some function of x, as X, or (D)y=X. If we know X and, we may seek that y=f(x) that will yield X on being subjected to the train of operations (D); i.e. we seek to invert at once the totality of operations (D), X so that y =-(D)X. This inversion $(D) is solving the Differential Equation (D)y=X, and is perhaps the most profound of mathematical operations, of immense and even unconquerable difficulty, overcome as yet only in day dy special cases. Thus x2 +4y=2x2, If we think of y2=ax2+2bx+c as a conic, and y-ẞ u(x-α) as a secant through a where &(D) =x2D2 −3xD+4, yields, as result point (a, 3) of the conic, then we may express y=x2(A+B log x+log x2), where A and B are arbitrary constants. Other forms of 4(D), quite as simple, yield far higher transcendents. Inverting a table of elementary D's we get a table of elementary Integrals. The art is to reduce other forms, if possible, to these elementary forms; when impossible, we must introduce transcendents defined by integrals. Change of Variable is the most fruitful method of Reduction. By mediate derivation, Dip(u) = '(u)• ux; also Dup(u) = '(u); L.C.M. of B, B', ... Herewith the III. Rational functions of x and both x and y rationally through u, which reduces this case to I; (a, B) may be taken variously. Generally we bring y2 to the form of sum or difference of two squares by putting x=z+h. IV. Rational functions of x and y, these being coördinates of a unicursal F(x, y) =o. We shall then have x=(u), y=4(u), where and are rational, whereby these Abelian Integrals reduce to I. The binomial x "(a+bxn)p can be reduced to II, and hence integrated, not generally, but in these important cases, by putting u=x" 1. integral; if (r and s integers), hences p'(u)du = p(u) = f(u)-uz-dx. In this put =z. of integration, multiply by the D of the old n = (rands integers), 3. +p integral; if p=(r and s integers), put a+bu=uz3. If f(x) be rational in x, Vax+b, Vcx+d, put ax+b=v2, which reduces it to 3. Of Transcendentals.—1. Rational functions of sin x and cos x. Put u=1 =tan a very impor tant substitution. χ 2 2. Rational functions of ear ̧ Put uea¤ ̧ 3. Rational Integral functions of x, eax ebx,... sin mx, sin nx, cos mx, cos nx, Express the sines and cosines through imagiexpress the Here imaginaries through sines and cosines. · Su,vdx+fuvzdx, or fuv.dx-uv-fuvdx. nary exponentials. In the result CALCULUS, THE INFINITESIMAL are included rational functions of the hypersine and -cosine. 4. Rational Integral functions of x and log x, or x and sin-1x. Put xe", or x = sin u. If f(x) =R(x, √T), T of 3d or 4th degree in x, cannot rationalize but must introduce Higher Transcendents. Let we dx ∞ dx VT ́4 (x — e1) (x — €2) (x — € 3) Herex and u are functions of each other and it seems natural to take u as function, x as dx argument; but in I =sin-1x, x= I-x2 sin I is a much simpler (periodic) function of I than I is of x; hence we may suspect that x above is a simpler (periodic?) function of u than u of x. Hence Abel thought the theory might be simplified by inverting the dependence before him assumed-one of the greatest divinations in mathematical history. We write dx x=(u) (Weierstrass), u = =8-1(x). Now just as sine and cosine have one period 27, SO has two periods, 20 and 2w', it is an Elliptic or Doubly periodic function. The Theory of such Functions, one of the most august creations of the last century, is conspicuous in Analysis. Of Hyper-elliptic Integrals there is no space to speak. The integral of an Infinite Series may be found by integrating term by term only when the series converges uniformly within an interval comprising the extremes of the integration. It is seen that the integrable forms are absolutely many, relatively few, the integration generally giving rise to a new function. Thus far we have raised no question as to Integrability, the Integrand being supposed unique, continuous, finite, and therefore integrable, in [a, b]. But when, if ever, may we let one or more of these conditions fall? As to continuity, Riemann has discussed profoundly, In what cases is a function integrable, and in what not? and still further precision has been attained by Du Bois Reymond and Weierstrass.* It is of particular interest to know whether y4x will vary finitely with varying modes of divisions of [a, b]. Riemann calls the sub-intervals d1, d2,...on; the greatest fluctuation of function-value in each ok he calls Dk; then must okDk be infinitesimal. Thence follows that when, as each dr sinks indefinitely toward o, the sum of sub-intervals, in which D is >o, itself is infinitesimal, then the Sum has a definite limit, the same how ever [a, b] be subdivided. Hence the integral I points in [a, b], and when f(x) has an ∞o of maxima and minima, or is quite undetermined (though finite) at a finite number of points in [a, b], as sin at 1 and 2; (x − 1)(x − 2) (3), even when f(x) is discontinuous or finitely indeterminate at an of points and has an ∞ of maxima and minima in the vicinity of an of points, provided only all these points of finite function-fluctuation form not a linear the function is then said to be only pointbut only a discrete mass of points (Punktmenge) wise (punktirt) discontinuous (Hankel).—Â dis or manifold of points is an of points in a finite interval [a−h, a+h], so distributed in subintervals that the sum of these subintervals may be made small at will by enlarging at will the number of subintervals. Otherwise, the mass is linear. Functions linearly discontinuous are not integrable. (For Cantor's more comprehensive theory of Derived Masses (Math. Ann., XVII, 358f), see Assemblages. crete mass Du Bois Reymond has shown ('Jour. f. d. r. u. d. a. Math., 79, 21f) that the product of two such integrable functions is itself integrable. Thus far the integrand has been finite. But the DI dx 1-x2 = at I. =/2, although the inte grand In general, if f(x) be ∞, or discontinuous, or oscillatory at x=c in [a, b], then f(x)dx loses meaning; but if the sum no matter how a and ẞ approach o indefinitely, then this limit is named value of ƒ ƒ(x)dx. Such is the case only when f(x)dx and f(x)dx converge each toward o, as a, a, B, ', all close down on o, a' <a, ß'<3; i.e. the immediate neighborhood of c must contribute infinitesimally to the integral. Similarly for any number of points not forming a linear mass. So, too, we may let either extreme, as b, increase towards, if only the total contribution of the infinitely remote region be infini tesimal, i.e. if f(x)dx <o for b however large, b being first taken sufficiently large. Double Integrals. ff(x, y)dx dy.—Think of a finite region R in XY, at each point of which is erected a perpendicular 2, all forming a cylindric volume (V) bounded by XY, the surface 1 = L £j(xk) dk = [] f(x)dx exists when f(x) = f(x, y), and the cylindric surface standing throughout [a, b]; but also (2) when f(x) is V=L2j(x,y)4R = ƒ f(x finitely discontinuous at a finite number of *For yet greater refinement and generalization, see Lebesgue. Annali di Matematica, 1902, 259f. f(x, y)dR. Here we assign no extremes to R, the integration stretches over all of R, so that B corresponds to the extremes of simple integration. It is and CALCULUS, THE INFINITESIMAL must be indifferent in what order the elements From DI we readily pass to the triple :, 4R are taken; hence we may sum frat such a fi(x, y, z)dx dy dz, and hence to mul a strip parallel to X, and then sum all Here for any value of x the values of y are determined by the equation of B. Hence b and are functions of x; but a and a depend on the extreme parallels to Y tangent to B, hence are absolute constants. a tiple I's in general. The higher Jacobian maintains its rôle: du do dw = ] dx dy dz, J = Vx Vy Vz ; and so on in general. Thus, to pass from rectangular to polar coordinates, x =pcos &, y=p sind; J(x,y; p, 0) =p; under ff, dx dy = pdød &—this latter is in curvilinear volumetric element under Analogy readily extends these forms to n-fold spaces. Thus the Jacobian appears geometrically as a real derivative, the limit of the ratio of two simultaneous changes. It is geometrically clear that II is perfectly fact the elementary curvilinear rectangle.-To definite, but we must ask in default of Geometry, pass from rectangular to spherical coordinates, when does approach the same limit index= cos sin o, y=p sin sin o, -p cos &, pendently of the function-value chosen for each whence J(x, y, z; p,, ) = p2 sin p, and AR and of the way in which each 4Ro, as their 2 sin pdp dedo is in fact the rectangular number∞o? Answer: When ΣDk 4Kko as each 4R=0, Dk being the greatest fluctuation in function-value in Rk. When is this the the case? Answer: (1) When f(x, y) is continuous throughout R; (2) when f at single points or on single lines (at points) becomes finitely discontinuous or indeterminate or oscillatory; (3) when f becomes thus finitely discontinuous or indeterminate oscillatory along an of lines (at 2 points), if only the sum of the elements (4R's), where D>o, is itself < (infinitesimal); i.e. when the or a •b The single ¡(x)dx=F(b)− F(a), where f=F' expresses the sum-value of f, integrated along the length b-a, through the end-values of linear masses do not form an areal (r planar) some Fat banda; can the double ƒ ƒ j(x, y) dx dy mass, i.e. when their initial elements form not a linear but only a discrete mass. May f(x, y) attain and II retain sense? Answer: Iff attains a definite, but only at definite points, or along a curve and of integrated over the region R also be expressed through the end-values of some F(x, y) along the contour of R? This query is much harder to answer, but is answered similarly: If f(x, y) order <1, then the II remains definite and is integrable in R, then in general f*¡(x, y)dx finite; also the order of integration remains indifferent. Here the contribution, to the II, o as the element of area (in XY) shrinks toward o along the curve; i.e. the volume V is for every included value of y a continuous function of x, F(x, y), and for every included value of x an integrable function of y. Then shoots up to only along an infinitely sharp theDI ffi(x, y)dx dy=fF(x, y) sin yds, where s is the contour of R and slope of the normal (drawn inwards at any point of s) to the + Y-axis. This latter is a curvilinear S, geometrically depicted as a wall built up Extremely important is the change of vari- (resp. down) along the contour s of R.— ables in II. In simple integration du f, whereby we pass from x under the uda Similarly a fff of f extended throughout to u a volume (or three-wayed spread) may be as variable of integration. In passing from expressed through the end-values of a certain F integrated over the entire surface (S) of a space-integral is turned into a surface-integral; and conversely: მ მა au av U x V x дх ду ду дх [Uy Vy] (u, v) =J(u, v; x, y). This remarkable d(x, y) expression, introduced by Jacobi and named by him Functional Determinant, is called Jacobian (Salmon). As already exemplified, it plays the rôle of derivative of the system ə(u, v) ə(x, y) (u, v) as to (x, y). In fact, (x, y) (u, v) ə (u, v) ở (w, z) (w, z) "d(x, y)' VOL. 4-7 CALCULUS, THE INFINITESIMAL common limit of the length of inscribed and circumscribed polygons of which each sideo. the integrand first, and then integrate, we Since st=√(x1)2+(yı)2, s=f√(xi)2+(yı)2dt. This holds generally, if for a definite interval not If now x=4(t), y=4(t), be continuous Volume is given by triple Integration, SSS dx dy dz, extremes defined by the bounding surface. Often the area of a section perpendicular (or possibly oblique) to an axis, as X, = [p, p+4p], and for [a, b], др (with possible exception of only discrete masses of points) a continuous function of both x and p; then მ 3 fi(x, др If p appears in either a or b or both, then a apf f(x, p)dx be an integrable function of p, F(p), in [a, B], integrating as top we get is a function of x, „S=j(x); then V-Sdx. f(x, p)dx' dp-f" F(p)dp, and the In the important case of Revolutes (of an area bounded by X, the curve, and two y-ordinates), zS =ny2, V=nfy2dx. Quadrature of a curved surface is sometimes called Complanation. Here again the area must be defined as the common limit of the surface area of polyhedra inscribed and circumscribed, no The surface element dS or (45) matter how. about P may be viewed as projected into the element dx dy in XY and as having a limiting ratio 1 with the corresponding element 4II in the plane tangent at P. The slope of this plane to XY=7=slope of normal to Z; AS 48 ΔΠ hence Lim. Ax Ay JI JxJy sec r={Fx2 + F y2 + F z2 } }/Fz; Lim. Among such the Eulerians are conspicuous, especially that of 2d species (Legendre), fx-le-dx, denoted by ľ(a), or better by the Gaussian II(a-1), through which countless others are expressible. This Ã- or II-function has remarkable properties: ff/Fdx dy, the region of integration in the factonal property, n=n\n=1. XY being the projection thereon of S, under For Revolutes, S=27 yds. CALCULUS, THE INFINITESIMAL a Cauchy's, Schlömilch's, come at once on apply. This ing the Maximum-Minimum Theorem. development holds under the two necessary and sufficient conditions (Pringsheim): 1. That p(x) possess everywhere in [x, x+R} definite finite differential coefficients of every finite order; 2. That Lim. · f(n) (xo+h)•kn converge uni().formly on o (for no) for all pairs (h, k) · for which oh≤h+k<R. (1+x)a+b dx, is denoted by B(a, b) (Binet) and is connected The Infinitesimal Analysis or Method of Limits is very highly developed and is applicable to almost every subject of exact thought, often asserting itself in the most surprising fashion, as in the Theories of Numbers and of Knots, to which it might seem wholly alien, suddenly unlocking and laying wide open secret .passages utterly unsuspected. In particular the Integral Calculus shows itself amazingly and unendingly fertile in the generation of new notions. As other and still other fields are exposed to investigative thought, the Calculus will receive more and more applications, and there seems to be no limit to the subtlety and refinement of its processes, to the keenness and penetration that may be given to this twoedged sword of the spirit, the strongest, sharpest, and most flexible ever fashioned or wielded by the mind of man. Historical Sketch.-Passing by anticipations, especially of Integration, that reach back at to Barrow's Lectiones opticæ et geometrica least to Archimedes (287-212 B.C.), we come (1669-70), on which Newton collaborated, how much no one knows. Barrow used the ƒ $′(y)dy=y&′(y) — ƒ 4′′ (yydy; -"(y)ydy (partially published first in Wallis's Works, vol. II, 1693) was shown to Barrow, Collins, Lord Brouncker in 1669, wherein he used o for a magnitude ultimately vanishing, as had James Gregory already in his (Geometriæ pars universalis (1667, Venice). Remainder); He treated Rectification, Cubature, and Mass-Center determinations as reducible to Quadrature and to be solved by introducing the notion of (Momentum)=instantaneous change, thus going beyond Barrow. Newton's Methodus Fluxionum et Serierum infinitarum' was 42 $(h) = p(0)+hp′(h) - - p'' (h) + —p''' (h) — ... 12 3 ́ynq(n+1)(y)dy. To avoid the alterna tion in sign we take | p(a−u)du and proceed as ready for the press before 1672, but not printed before: then on putting a=x,+h, 4(x+h) = f(x)+hp'(xo)+ + I till 1736. In it he proposes, (1) to find the velocity at any instant from the space traversed up to each instant, (2) to find the latter (space) from the former (velocity)-the two problems of Derivation and Integration conceived kinematically. The equicrescent mag funqn+1(x+h−u)du, o<h<R. nitude x, as a space, is called fluens (Cavalieri Such is the swiftest, directest, nearest-lying deduction of the fundamental Taylor's Series, by which the value of at (x+h) is built up out of the value of and its D's at xo. The Rn is here yielded as a definite integral, from which form the other forms, as Lagrange's, fluens, 1639, Napier fluxus, 1614. Clavius fluere, 1574); the velocity he writes and calls fluxio-our Derivative (as to the time t). Momentum varies as fluxion, is written xo, and corresponds to our Differential x (incrementa indefinite parva). This treatise seems to have been revised after 1673, hence |