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Mathematics - Schaum%27S Outline Of Theory And Problems Of Advanced Calculus 2

(BQ) Part 2 book "Advanced calculus" has contents: Multiple integrals; line integrals, surface integrals, and integral theorems; infinite series; improper integrals, fourier series, fourier integrals, gamma and beta functions, functions of a complex variable. | Multiple Integrals Much of the procedure for double and triple integrals may be thought of as a reversal of partial differentiation and otherwise is analogous to that for single integrals. However, one complexity that must be addressed relates to the domain of definition. With single integrals, the functions of one variable were defined on intervals of real numbers. Thus, the integrals only depended on the properties of the functions. The integrands of double and triple integrals are functions of two and three variables, respectively, and as such are defined on two- and three-dimensional regions. These regions have a flexibility in shape not possible in the single-variable cases. For example, with functions of two variables, and the corresponding double integrals, rectangular Fig. 9-1 regions, a @ x @ b, c @ y @ d are common. However, in many problems the domains are regions bound above and below by segments of plane curves. In the case of functions of three variables, and the corresponding triple integrals other than the regions a @ x @ b; c @ y @ d; e @ z @ f , there are those bound above and below by portions of surfaces. In very special cases, double and triple integrals can be directly evaluated. However, the systematic technique of iterated integration is the usual procedure. It is here that the reversal of partial differentiation comes into play. Definitions of double and triple integrals are given below. Also, the method of iterated integration is described. DOUBLE INTEGRALS Let Fðx; yÞ be defined in a closed region r of the xy plane (see Fig. 9-1). Subdivide r into n subregions Árk of area ÁAk , k ¼ 1; 2; . . . ; n. Let ð k ; k Þ be some point of ÁAk . Form the sum n X Fð k ; k Þ ÁAk ð1Þ k¼1 Consider lim n!1 n X Fð k ; k Þ ÁAk k¼1 207 Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ð2Þ 208 MULTIPLE INTEGRALS [CHAP. 9 where the limit is taken so that the number n of subdivisions increases without limit and .