Advanced Calculus fi..

Chapter 4 Integral Calculus of Functions of Several Variables 231The following property of double integrals is established as for single integrals(Problems 1 1 and 12 following Section 4.1 ; see also Problem 14 following Section2.18):Let f (x, y) be continuous in the bounded closed region R. Iffor every bounded closed region RI contained in R, then f (x, y) E 0 in R. Iff (x, y) L 0 in R andthen f (x, y) = 0 in R.Remarks on the definition of the double integral. For a subdivision such as that ofFig. 4.3, one can include in the sum (4.29) a term for each rectangle partly includedin R, using a point (xf, y:) of R in the rectangle and taking as A, A the area of theirregular piece. For the type of region considered here, the total contribution of suchterms can be shown to be negligible in the sense that for h sufficiently small, theirsum has an absolute value that is less than a prescribed positive 6. The same assertionholds if one includes the boundary pieces and in each case uses as Ai A the area ofthc whole rectangle.One can also subdivide R by curves other than straight lines, as in Fig. 4.6.Under some natural restrictions on these curves and on R the areas A I A, A2A, . . .are well determined, and the sum (4.29) can be formed. The mesh h is now themaximum "diameter" of the n pieces (see Fig. 4.6). If f is continuous in R, thenEq. (4.30) remains valid. Because of the freedom to use arbitrary shapes, one alsowrites the double integral asFigure 4.6General subdivision for a double integral.