Advanced Calculus fi..

320 Advanced Calculus, Fifth Edition 3 {-jb) 11,x2 dy dz + 2xy dz dx + 2x2 dx dy = 6V.f. jwhere (f, j, 2) is the centroid of R;c) j fs curl v . n da = 0, where v is an arbitrary vector field. 13. Deduce the results of Problem 9(a), (b), (c) following Section 5.10 by proving (c) first, 4using the incompressibility of the flow of constant velocity v.4. Let S be the boundary surface of a region R, with outer normal n, as in the Divergencetheorem. Let f (x, y, z) and g(x, y, z) be functions defined and continuous, with continuousfirst and second derivatives, in a domain D containing R. Prove the following relations:a) ffsf agian do = jfif v2g d~ dy dz + j&(vf. vg)d~ dy dz;[Hint: use the identity V . (f u) = Vf - u + f (V. u).]b) if g is harmonic in D, then[Hint: Put f = 1 in (a).]c) iff is harmonic in D, thend) if f is harmonic in D and f = 0 on S, then f E 0 in R [cf. the last paragraph beforethe remarks at the end of Section 4.31;e) iff and g are harmonic in D and f = g on S, then f = g in R; [Hint: Use (d).]1f) iff is harmonic in D and af/an = 0 on S, then f is constant in R;g) if f and g are harmonic in D and aflan = aglan on S, then f = g + const in R;h) if f and g are harmonic in R, andthenf E g in R;i) if f and g both satisfy the same Poisson equation in R,v2f = -4nh, v~~ = -4nh, h =h(x, y,~),and f = g on S, thenj) jh (f 8 - i g)do = ffjR(f v 2f ~ ginR;- gv2f)dx dy dz;[Hint: Use (a).] 4k) if f and g are harmonic in R, then