Du Bois Reymond has shown f. d. r. u. d. a. Math.,' 79, 21f) that the product of two such integrable functions is itself inte grable.
Thus far the integrand has been finite. But the DIf 1 dx = r/2, although the inte grand= 00 at 1. In general, if f(x) be OD or dis continuous, or oscillatory at .r==c in [a, b], then fb i(x)dx loses meaning; but if the sum (x)dx- F f i(x)dx nears a definite limit e +0 no matter how a and ft approach 0 indefinitely, b then this limit is named value of f f (x)dx.
a a--a' Such is the case only when f f(x)dx and converge each toward 0, as a, a', fi, pi, all close down on 0, a' < a, 13' < P; i.e., the immediate neighborhood of c must con tribute infinitesimally to the integral. Simi larly for any number of points not forming a linear mass.
So, too, we may let either extreme, as b, increase toward ao if only the total contribu tion of the infinitely remote region be infini b tesimal, i.e., (x) dx
Double Integrals. iff(x,y)dady.—Think of a finite region R in XY, at each point of which is erected a perpendicular z, all forming a dric volume (V) bounded by XY, the surface se=f(x, y), and the cylindric surface standing on the border (B) of R. To find V we may cut up R into elements (JR), as by parallels to X and Y, and V into elementary cylinders (AV) on these elementary bases. Plainly y) JR-- ff (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 must be indifferent in what order the elements JR are taken; hence we may sum up first along a strip parallel to X, and then sum all such strips along Y. This double summing is ex pressed by a Double Integral (II) thus : LZ[Ef (or, a f (x, y) dx dy.
a b Here for any value of x the values of y are determined by the equation of B. Hence b and 6 are functions of x; but a and a depend on the extreme parallels to Y tangent to B, hence are absolute constants.
It is geometrically clear that II is perfectly definite, but we must ask in default of Geometry, when does 2 approach the same limit inde pendently of the function-value chosen for each JR and the way in which each JR 0, as their number.-- 03 ? Answer: When E Dk•,IRk=_.- 0 as
each 111?-0, Dk being the greatest fluctuation in function-value in dRk. When is this the case? Answer : (1) When f(x, y) is con tinuous throughout R: (2) when f at single points or on single lines (at points) be comes finitely discontinuous or indeterminate or oscillatory; (3) when f becomes thus i finitely discontinuous or or oscillatory along an 00 of lines (at points), if only the sum of the elements (JR's), where D>a, is itself (infinitesimal) ; i.e., when the linear masses do not form an areal (or planar) mass, i.e., when their initial elements form not a linear but only a discrete mass.
May f(x, y) attain 00 and II retain sense? Answer: If f attains a definite 00 but only at definite points, or along a curve and of order <1, then the II remains definite and finite; also the order of integration remains indifferent. Here the contribution, to the //, =-:-_ 0 as the element of area (in XY) shrinks toward 0 along the curve; i.e., the volume V shoots up to 00 only along an infinitely sharp edge.
So, too, the region R may stretch out any way toward 00 if f shrinks faster than R spreads ; e.g., R may spread over all the plane, if in all remote regions f becomes 0 of higher than 2d order. Minuter discussion must be foregone.
Extremely important is the change of vari ables in II. In simple integration du=u,.dx under the f , whereby we pass from x to u as variable of integration. In passing from x, y to U, v, under theff ,du (17.M dx dy, but 8u 8vauov lux v what is M? It is — • — — — — = ax ay ay' ax a(u, v.) l(u, v; .r, y)• This remarkable a(x, Y) expression, introduced by Jacobi and named by him the Functional Determinant, is called the Jacobian (Salmon). As already exemplified, it plays the role of derivative of the system (u,v)asto(x,y),InfactMV=1 dx -= , a(u, v) a(u, v) a(tv,z) aS — — 1.