18.02a Problem Set 11(due Thurs., Jan. 24)Part I (15 points)TB = textbook; SN = Supplementary Notes (all have solutions)Topic 44 (Thur., Jan. 17, pset 10 due) Surface integrals.Read: SN: V8, V9Hand in: 6A/1, 3, 4; 6B/1, 2, 3, 4, 5, 8Others: 6A/2; 6B/6Topic 45 (Tue., Jan. 22) More surface integrals, divergence theorem.Read: SN: V10Hand in: 6C/3, 5, 7a, 8Others: 6C/1a, 2, 6Topic 46 (Wed., Jan. 23) Applications and interpretations of divergence theorem.Read: SN: V15 sec. 1 for div in ∇ notation, V15 sec.2 to middle p.3Hand in: 6C/10, 11Topic 47 (Thur., Jan. 24, pset 11 due) Line integrals in 3D. Conservative vector fields.Read: SN: V11, V12Topic 48 (Fri., Jan. 25) Stokes’ theorem.Read: SN: V13Part II (27 points)Problem 1 (Topic 43, 2 pts)An attractive particle know as the Perepelitsa (P for short) place d at the origin createsa radially force field with magnitude equal to the distance from the origin. A negative Pcreates a field of the same magnitude and the oppositie direction.A positive P is placed at (0, 0, −1) and a negative one at (0, 0, 1). Compute the resultingforce field.Problem 2 (Topic 45, 8 pts:1,3,2,2)Consider a tetrahedron with vertices A = (0, 0, 0), B = (1, 0, 1), C = (1, 0, −1), and D =(1, 1, 0).a) Which faces are exchanged by the symmetry z → −z?b) Find outward pointing normals to each face (find the easiest ones, not necessarily unitnormals). (Note: two of the faces have easy normals, you can calculate one of the normalsand use symmetry to find the last one.)c) Let F = y i. Calculate the flux of F through each face. (For one of the faces you willwant to use the formula n dS =Nk·Ndx dy given in the notes. Again, you can use symmetryto avoid one calculation.)d) Verify the divergence theorem for the tetrahedron and the vector field F by computingeach side of the formula.(continued)118.02a Problem Set 11 2Problem 3 (Topic 44, 3 pts)Consider a solid in the shape of an ice-cream cone. It’s bounded above by(part of) a sphere of radius a centered at the origin. It’s bounded belowby the cone with vertex at the origin, vertex angle 2φ0and slant heighta.Find the gravitational force on a unit test mass placed at the origin.(Assume density = 1.)aφ0Problem 4 (Topic 45, 7 pts:2,3,2)Continuing with the solid in problem 3: take a =√2 and the vertex angle to be π/2 (soφ0= π/4). Let F = zk.a) Let T be the horizontal disk with boundary the intersection of the sphere and the cone.Compute directly the upward flux of F through Tb) Let U be the boundary of the conical lower surface, S the upper spherical cap of thesolid. Use the divergence theorem and part (b) to compute the upward flux of F throughU and S. (You will need to be careful with signs.)c) Set up, but don’t compute, the integral for the flux of F through U. Write the integralin cylindrical coordinates.Problem 5 (Topic 45, 7 pts:2,2,3)Let f(x, y, z) = 1/ρ.a) Compute F = ∇f and show divF = 0.b) Find the outward flux of F through the sphere of radius a centered at the origin. Whydoes this not contradict the divergence theorem?c) Analogous to Green’s theorem, the divergence theorem can be extended to the casewhen the boundary of the solid volume consists of more than one closed surface. For eachboundary surface the outward normal is the one pointing away from the solid.Imititating what we did with Green’s theorem, show that the flux of F through any closedsurface surrounding the origin is
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