JS = JS JII JII Hence Lim. — Jx4y JIT sec Fr' F5' i/F,; hence S } 4/F,dx dy, the region of integration in XY being the projection thereon of S, under obvious conditions. Often the surface is given parametrically, i.e., x, y and z as func tions of the independents, u and v. Then we write off the rectangular array II x4x, II and form therefrom the three Jacobians, by deleting the columns in order, commonly written A, B, C. Hence, by easy substitution for Fp, F,, S =ff V il' + B' + C' du dv'. To every point (x, y, z) of S there is co-ordi nated a point (u, v) of a plane.
For Revolutes, yds.
Differentiado de Curva in We have found the D of anf as to either extreme, but the integrand may contain a parameter, thus ; p)dx. being then a function of f, is also a function of p and may be differen tiated as to p, giving rise to Parametric Deriva tion. Thus, /=-f epzdx =-- ep. r • ' hence a a tox__ 1 b /p=- I ePx . If now we differentiate a the integrand first, and then integrate, we obtain the same result. Hence the order Integr. x Der. p is equivalent to Der.p Integr. x This holds generally, if for a definite interval (x. 0 [p, p- f- JpI, and for la, b], is in general (with possible exception of only discrete masses of points) a continuous function of a b both x and p; then— f f(x, dx. OP a ap If p appears in either a or b or both, then a dd da fbal(x, p) =i(k P) p) d+ a ap dx.
6 If the i be an integrable function of p, a F(p), in [a, ti], integrating as to p we get f a p --- f E(p)dp, and the a order of integration is indifferent while f is a continuous function of both x and p.
b Thus themay be treated as function of i a any parameter in f, when the extremes are constant, as 0, 1,
cc' , and this gives rise to an important set of concepts and to the Theory of Definite Integrals. Thus, for
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discovery. Hereby arguments complemental to 1 are set in mutual relation, as sin ( - a) = sin a, tan a•tan 7
so that from F(I) we may reckon F(i). Hence we need reckon f(a) only for a in [0, I], as sin a only for a in [0, r/2].
r(a) and particularly log r(a) may now be differentiated with highly interesting results, log F(a)ee=-'n 1 as dal . This E is uni - n=o formerly convergent for a>0; hence we may integrate term-wise from 1 to a and get s,—_00 d log (a - 1) 1 where da (1+n) (a+n) C is the Eulerian or Mascheronian Constant ... calculated by Euler to 15 and by Legendre to 19 decimals. Hence, by a 2d Integration, m log F(a)=-- log n1 a+m-1 ' ms=coM (-- r(a)=II • m=l a +m- 1 m(m+1)"-1 a(1+a) (1+ ) (m+1)a , (in=-*_.0e ).
I m- a a(1 +a) (1, 2 . . . (1+ Such is Gauss's Definition of r(a) for every finite a for which no factor in the denominator vanishes. Herewith we are brought' to the expression of functions not through infinite series but through infinite products, as already exemplified in Wallis's formula: w 2 2 4 4 6 6 2 1 3 3 5 5 7 • This subject, of infinite range, cannot be pursued here -"hills peep o'er hills, and Alps on Alps arisen The fundamental theorems were rigorously proved first by Weierstrass (Jour. f. d. r. u. d. a. Math., LI). It may be added that the 1st Eulerian, 0 (1+x)a+b is denoted by B(a, b) (Binet) and is connected with the 2d by and being expressible thus simply through r has not so much independent significance.
The central notion of the Integral Calculus, the LIMIT of a Sum, is more obvious than that of the Differential Calculus, the LIMIT of a QUOTIENT. The foundations of the one are also seen to be much broader than those of the other, so that the former is not merely the in verse of the latter. The twain seize upon the two great aspects of History, the Dynamical and the Statical, Process and Result. While the Integral Calculus borrows its speed and direct ness from the Differential Calculus, its own reaction upon this latter is instant and power ful. Thus, from integration'by parts we have - f ib"(Y)YdY 11 - 2 y'01(y)+ 12 - and so on.