Question

(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain the origin, it is in the domain of F, so you can apply the divergence theorem closed surface in R3 that encloses the origin. We will show that the (d) Now suppose that S is a flux of F through S equals 47T (i) Let R be small enough so that the sphere SR of radius R is entirely contained in the interior of S. If we parametrize Sr using spherical coordinates, then the normal vector is N R2 sin o (cos 0 sin ø, sin0 sin ø, cos ø). Show that F N = sino. (ii) Show that the flux of F through Sr equals 47 (iii) Let W be the solid 3D region between S and SR. Convince yourself that the boundary of W is S SR (ie. S is oriented with outward-pointing normals, while SR is oriented with inward-pointing normals). (iv) Show that the flux of F through the boundary of W is zero. Hint: since W does not contain the origin, you can apply the divergence theorem. (v) Show that the flux of F through S equals 4T. R3 \ {(0,0,0)} such that F curl(A)? (e) Does F have a vector potential A defined on Hint: suppose F = curl(A). What does Stokes' Theorem tell you about the flux of F through any closed surface S in R3?

Answer 1

F = (x, y, z)/r^3 = x_i + y_j + z_k/r^3 = x/r^3 i + y/r^3 j + z/r^3 k = F_1 i + F_2 j + F_3 k Implies F_1 = x/r^3 we have d (u/v) = vu' - uv'/v^2 a) Now partial differential F_1/partial differential x = r^3 middot (1) middot 3r^2 middot partial differential r/partial differential x /(r^3)^2 = r^3 - 3 x r^2 partial differential r/partial differential x/r^6 we have r = squareroot x^2 + y^2 + z^2 Now partial differential r/partial differential x = 1/2 squareroot x^2 + y^2 + z^2 middot 2x = x/ r Therefore partial differential F_1/partial differential x = r^3 - 3 x r^2 middot (x/r)/r^6 = r^3 - 3 x^2 middot r/x^6 = r^2 - 3 x^2/r^5 b) div (F) = nabla middot F = partial differential F_1/partial differential x + partial differential F_2/partial differential y + partial differential F_3/partial differential z^2 = r^2 - 3 x^2/r^5 + r^2 - 3 y^2/r^5 + r^2 - 3 z^2