Advanced Calculus fi..

286 Advanced Calculus, Fifth Edition4 The integral $, P dx + Q dy can also be interpreted, as above, as the integralfv - n ds = fun ds,Cwhere v is the vector Qi - Pj. The right-hand member of Green's theorem is thenthe double integral of div v. Thus one has, for an arbitrary vector field,Cwhere n is the outer normal on C, as in Fig. 5.6(a). This result is the 2-dimensionalform of Gauss's theorem, to be considered later.Application to area.Equation (5.24) asserts thatThis follows at once from Green's theorem sinceIf these two equal line integrals are averaged, another expression for area is obtained:This can also be checked by Green's theorem.0PROBLEMS1. If v = (x2 + y2)i + (2xy)j, evaluate 1, v~ ds for the following paths:a) from (0,O) to (1, 1) on the line y = x,b) from (0,O) to (1, 1) on the line y = x2;c) from (0,O) to (1, 1) on the broken line with corner at (1,O).2. Evaluate lc un ds for the vector v given in Problem 1 on the paths (a), (b), and (c) ofProblem 1, n being chosen as the normal 90" behind T.3. The gravitational force near a point on the earth's surface is represented approximatelyby the vector -mgj, where the y axis points upwards. Show that the work done by thisforce on a body moving in a vertical plane from height h to height h2 along any path ,isequal to mg(hl - h2).4. Show that the earth's gravitational potential U = -kMm/r is equal to the negative of thework done by the gravitational force F = -(k~m/r~)(r/r) in bringing the particle to itspresent position from infinite distance along the ray through the earth's center.-,?