MATH 251H Final Exam Version B Solutions

2 2 25 points verify gauss theorem f dv f ds v v for

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Unformatted text preview: 2 4 sin 2  z2 3  4  1  9  9 M  ÞÞÞ  dV  Þ Þ Þ r 3 sin  cos  z dr d dz  r 40 2 20 22 00 0 0 /2 3 2 3 /2 2 5 2 sin 3  z M xz  ÞÞÞ y dV  Þ Þ Þ r 4 sin 2  cos  z dr d dz  r  32  1  9  48 50 5 5 3 20 32 00 0 0 M xz  48 1  16 y M 59 15 3 15. /2 2 (25 points) Verify Gauss’ Theorem   ÞÞÞ   F dV  ÞÞ F  dS V V  for the vector field F  2xy 2 , 2yx 2 , z x 2  y 2 above the cone z  2 x 2  y 2 and the solid below the plane z  2. Be careful with orientations. Use the following steps: First the Left Hand Side: a. Compute the divergence:   F  2y 2  2x 2  x 2  y 2  3x 2  3y 2  b. Express the divergence and the volume element in the appropriate coordinate system:   F  3r 2  dV  r dr d dz c. Find the limits of integration: 0  2 2 x2  y2 z 2 becomes 2r z 2 2r  2 when r  1 d. Compute the left hand side: 2 1 2 1 1  ÞÞÞ   F dV  Þ Þ Þ 3r 2 r dz dr d  2 Þ 3r 3 z 2 dr  6 Þ r 3 2 2r dr 0 V  12 Þ r 3 1 0 0 0 2r 4 r 4 dr  12 r 4 r5 5 1 0 2r  12 1 4 1 5 0  3 5 5 Second the Right Hand Side: The boundary surface consists of a cone C and a disk D with appropriate orientations. e. Complete the parametrization of the cone C:  R r,   r cos , r sin , 2r f. Compute the tangent vectors: r  cos , sin , 2 e   r sin , r cos , 0 e g. Compute the norma...
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This note was uploaded on 02/06/2014 for the course MATH 251H taught by Professor Philipb.yasskin during the Spring '11 term at Texas A&M.

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