MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Problem Set 11 Due Thursday 11 29 07 12 45 pm 17 points Part A Hand in the underlined problems only the others are for more practice Lecture 30 Thu Nov 15 Vector elds in 3D surface integrals and ux Read Notes V8 V9 Work 6A 1 2 3 4 6B 1 2 3 4 6 8 Lecture 31 Fri Nov 16 Divergence theorem Read Notes V10 15 6 Work 6C 1a 2 3 5 6 7a 8 Lecture 32 Tue Nov 20 Divergence theorem continued applications and proof Read Notes V10 15 6 and pp 1054 1055 about heat ow Lecture 33 Tue Nov 27 Line integrals in space curl exactness and potentials Read Notes V11 V12 p 1017 1018 on curl Work 6D 1abcd 2 4 5 6E 1 2 3ab ii both methods 5 26 points Part B Directions Attempt to solve each part of each problem yourself If you collaborate solutions must be written up independently It is illegal to consult materials from previous semesters With each problem is the day it can be done Write the names of all the people you consulted or with whom you collaborated and the resources you used Problem 1 Thursday 11 15 2 points Notes 6B 7 Problem 2 Friday 11 16 8 points 1 2 3 2 Consider a tetrahedron with vertices at P0 0 0 0 P1 1 0 1 P2 1 0 1 and P3 1 1 0 a Which two faces are exchanged by the symmetry z z b Find normals to each face use the easiest ones not unit normals pointing outwards You can avoid one calculation using symmetry c Calculate the ux of F y through each face Because F is symmetric with respect to z z you can again avoid one calculation using symmetry d Verify the divergence theorem for the tetrahedron and the vector eld F by computing each side of the formula Problem 3 Friday 11 16 5 points 2 1 2 a Let f x y z 1 x2 y 2 z 2 1 2 Calculate F f and describe geometri cally the vector eld F b Evaluate the ux of F over the sphere of radius a centered at the origin c Show that div F 0 Does the answer obtained in b contradict the divergence theorem Explain 1 Problem 4 Tuesday 11 20 6 points 1 2 2 1 The Laplacian of a function f is the quantity 2 f fxx fyy fzz a function f is harmonic if it satis es the Laplace equation 2 f 0 a Show that if S is a closed surface bounding a region D and if f has continuous second derivatives inside D then f n dS 2 f dV In particular the ux of S D the gradient of a harmonic function through a closed surface is always zero b Show that if S is a closed surface bounding a region D and if f has continuous second derivatives inside D then f f n dS f 2 f f 2 dV S D c Use the result of b to show that if a function f is harmonic everywhere in D and f is zero at every point of the boundary S then f is zero everywhere in D rst show that f 0 d Deduce that if two functions f and g are harmonic everywhere in D and f g on the boundary S then f g everywhere in D Problem 5 Tuesday 11 27 5 points 2 1 2 a Compute in terms of the constants a b the work done by the vector eld F a sin z bxy 2 2x2 y x cos z z 2 k along the portion of helix x cos t y sin t z t from 1 0 0 to 1 0 2 b Compute curl F For which value s of a and b is the vector eld F conservative c Let a and b be the values you found above Find a potential function for F using a systematic method and verify the answer you found in part a using the fundamental theorem of calculus 2
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