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296 THE TRANSFORMATION OF MULTIPLE INTEGRALS. [216.
THE TRANS FORMATION OF
Or, if the given integral is
dx, then if we replace x and y by af and bn respectively, and substitute for & and n in terms of u and v, the integral becomes abf (u) u dudv.
216.] Although the limits of the first integration of a multiple integral are functions of one or more of the variables, in reference to which subsequent integrations are to be effected, yet it is possible, and frequently convenient, so to change the variables, that the limits of the first integration may be constant. The mode of effecting this transformation is explained in Art. 93. Let us take the case of a double integral; and suppose it be 1 ="I"f(x,y) dy dx,
(60) where y, and y, are functions of x. Let y = yo+(Yı-yo)t,
(61) where t is a new variable; therefore
dy = (yı-yo) dt; and observing that t =1, when y=yı; and t = 0, when y=yo; we have
217.] DIFFERENTIATION OF DEFINITE INTEGRALS. But / e-(22+b+c) ** dx = - 1 pente
2(a2 + 62 12) 4.0
integral oral, due biect
SECTION 3.—The Differentiation of a Definite Multiple Integral
with respect to a Variable Parameter. 217.] If a parameter, capable of variation, is contained in the element-function of a definite multiple integral, and also in some or all of the limits of integration, the integral is a function of that parameter; so that the value of the integral will change, if the parameter varies. Let us suppose the parameter continuously to vary, within such limits however that the element-function of the integral does not become infinite or discontinuous; then the value of the integral will also continuously vary; and it is the increment of the integral, due to the infinitesimal variation of the parameter, which it is our object to determine in the present section.
In Art. 96, we have investigated the change of value of a definite single integral, due to the infinitesimal variation of a parameter, of which the limits as well as the element-function are functions; and we have shewn that if a is the variable parameter, and D denotes the total differential, of(a,x)dx=P(a,x») dx – "(0,) dx+ ** d.F. 4;4) dadx ; (63)
(a, x) dx =
Wao Tao da Now this will be the foundation of our subsequent investigations. Let us first consider a definite double integral; viz.
F(a, x, y) dy dx = | in dy dt, (65) where
n = f(a, x, y); PRICE, VOL. II.
and where Xn, X, Yn, Yo, as also , are all functions of a; a being a parameter capable of continuous variation. We have to determine the change of value in u due to an infinitesimal variation of a.
In (64) let f(a, x) be replaced by /r (a, x, y) dy; that is, by In dy; then
and in the first term of the right-hand member of (67), inasmuch
r un dx . 7*.
which gives the total variation of the definite double integral.
Again, if the similar variation of a definite triple integral is required, let a in (68) be replaced by l odz, where the new a is P(a, x,,z); then by an exactly similar process it may be shewn that PL/" An dz dy da
dx. (69) Xo'yo Laad 20
Jao Jyo J'zo da Now the law of formation of the several terms in the second member of this equation is sufficiently obvious, so that the variation of a definite multiple integral may be easily expressed at length; and the general law may be proved inductively; viz. by assuming the truth of it for a multiple integral of the (n-1)th order, and deducing from that the same law for the multiple of the nth order. This process is so easy that it is unnecessary to express it at length.
218.] Definite multiple integrals also, when one integration has been effected, and the limits of that integration are functions of a variable parameter, are subject to variation by reason of the change of that variable parameter. The consequent variable of the reduced integral may be derived from the theorem given in (64). The following are examples of the process.
In (64) let F (a, a) be replaced by "; then