Advanced Calculus fi..

352 Advanced Calculus, Fifth EditionFigure 5.'d7 Formation of double layer.and, since R log 1 / R + 0 as R + 0, the limit does exist. The conclusion also holdsfor p of variable sign (as occurs in the electrostatic case) (see Section 6.26).Obtaining F from U requires differentiating under the integral sign. For Poutside E, Eq. (5.164) follows from Leibnitz's Rule of Section 4.9. For P in E theresult is correct, but a more careful analysis is needed. One can in fact show thatgrad U is given by (5.164) for all (x, y) and both U and grad U are continuouseverywhere.One can also, by a passage to the limit from (5.162), obtain the logarithmicpotential of a distribution of mass on a curve C. One is led to a line integralHere we find that if C is piecewise smooth and p is continuous, then U is continuouseverywhere. However, the corresponding force field grad U is defined only for Poff of C.Another limit process starts with two "parallel" curves C,, C2, close togetherwith mass (or charge) densities equal but opposite in sign at adjacent points, assuggested in Fig. 5.40. If the two curves are brought to coincidence while the densitiesbecome infinite in the proper manner, one obtains as limit a new potentiala 1U(X, Y) = 1 Y(s)- an log - r ds, (5.167)Ccalled the logarithmic potential of a double layer on C. In electrostatics this correspondsto a dipole layer on C. The curve C is assumed to have a continuously varyingunit normal vector n, with respect to which the directional derivative alan is formed;y(s) is assumed to be continuous on C. By contrast to (5.167) the potential (5.166)is called the potential of a single layer on C. For (5.167), one finds that U itself isdiscontinuous on C. However, for P off of C, U is well defined and continuous, andLeibnitz's Rule can be applied to form the force field F, which is also continuousoff of C.4EXAMPLE The logarithmic potential of a mass distribution of constant densityb over a disk of radius a with center at (0,O) is given by'I = b la lLllog 1 p dB dp. (5.168)R2 + p2 - 2Rp COS(~ - ff)Here x = R cos a, y = R sin a, and we used the law of cosines to express r interms of R, p, 8, a, as in Fig. 5.41. With the help of advanced integration formulas,