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 V9 Hand in 6A 1 3 4 6B 1 2 3 4 5 8 Others 6A 2 6B 6 Topic 45 Tue Jan 22 More surface integrals divergence theorem Read SN V10 Hand in 6C 3 5 7a 8 Others 6C 1a 2 6 Topic 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 3 Hand in 6C 10 11 Topic 47 Thur Jan 24 pset 11 due Line integrals in 3D Conservative vector fields Read SN V11 V12 Topic 48 Fri Jan 25 Stokes theorem Read SN V13 Part II 27 points Problem 1 Topic 43 2 pts An attractive particle know as the Perepelitsa P for short placed at the origin creates a radially force field with magnitude equal to the distance from the origin A negative P creates 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 resulting force 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 unit normals Note two of the faces have easy normals you can calculate one of the normals and 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 will N want to use the formula n dS k N dx dy given in the notes Again you can use symmetry to avoid one calculation d Verify the divergence theorem for the tetrahedron and the vector field F by computing each side of the formula continued 1 18 02a Problem Set 11 Problem 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 below by the cone with vertex at the origin vertex angle 2 0 and slant height a 2 0 Find the gravitational force on a unit test mass placed at the origin Assume density 1 Problem 4 Topic 45 7 pts 2 3 2 Continuing with the solid in problem 3 take a 2 and the vertex angle to be 2 so 0 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 T b Let U be the boundary of the conical lower surface S the upper spherical cap of the solid Use the divergence theorem and part b to compute the upward flux of F through U 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 integral in cylindrical coordinates Problem 5 Topic 45 7 pts 2 2 3 Let f x y z 1 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 Why does this not contradict the divergence theorem c Analogous to Green s theorem the divergence theorem can be extended to the case when the boundary of the solid volume consists of more than one closed surface For each boundary 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 closed surface surrounding the origin is 4 a
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