Advanced Calculus fi..

4-354 Advanced Calculus, Fifth EditionThis is illustrated by the example of Eq. (5.168), for which U is a constant times ?!log 1/R (R2 = x2 + y2), away from the mass.Behavior of U and F for large R. The results to be presented hold for all types ofmass distributions, but we formulate them only for the case of a continuous massdistribution on E, as in (5.163). We ask how U and F behave when P: (x, y) recedesto infinite distance, that is, when R + oo, R = (x2 + y2)1/2. We letBREwe interpret M as the total mass.For U itself we have the following assertion: For large R, R 2 Ro,where p(x, y) is bounded, Ip(x, y)l 5 const. Thus for large R the potential behavesas if the mass were concentrated at the origin, with a small error, which approachesOasR-+ oo.To prove this, we led d be so large that E is included in the circular regionx2 + y2 j d2. Then by the triangle inequality (a + b 2 c for a triangle of sidesa, b, c),(see Fig. 5.43). Therefore for R > d,R~r+d, r ~ R + d,.so thatFigure 5.43 Derivation of Eq. (5.172).