Advanced Calculus fi..

284 Advanced Calculus, Fifth EditionFigure 5.12Decomposition of region into special regions.In the same way, JJR 2 dx dy can be written as an iterated integral, with the aid of(5.42), and one concludes thatAdding the two double integrals, one finds: ,$I/ (g- E )dxdy =fPdx+ Qdy. f 3RThe theorem is thus proved for the special type of region R.Suppose next that R is not itself of this form but can be decomposed into a finitenumber of such regions: R1, R2, . . . , R, by suitable lines or arcs, as in Fig. 5.12.Let C1, C2, . . . , C, denote the corresponding boundaries. Then Eq. (5.40) can beapplied to each region separately. Adding, one obtains the equation:$(pdx+ady)+ ( )+.- 1ClfC2=/I(%-%)R IBut the sum of the integrals on the left is justCc,, fax ay dxdy+...+/J( )dxdy. 3R,,'iFor the integrals along the added arcs are taken once in each direction and hencecancel each other; the remaining integrals add up to precisely the integral around Cin the positive direction. The integrals on the right add up to