Chapter 1. calc_tutorial_17_3_013

1.1 Problem Statement

{5}
{2}
pow($b,2)
3*$a
$d/5
32*$a/5

Use the Divergence Theorem to evaluate the flux .

F = <$ax3, 0, $az3>, S is the octant of the sphere x2 + y2 + z2 = $c, in the first octant x ≥ 0, y ≥ 0, z ≥ 0.

1.2 Step 1

Question Sequence

Question 1.1

Recall how to find the divergence of a vector field F = <F1, F2, F3>.

div(F) = (∂F1 / ∂x) + (∂F2 / ∂y) + (∂F3 / ∂z)

Recall the Divergence Theorem. Let S be a closed surface that encloses a region W in R3. Assume that S is piecewise smooth and is oriented by normal vectors pointing to the outside of W. Let F be a vector field whose domain contains W. Then we have the following equation.

To evaluate the flux, we will need to find the divergence of the vector field F and then use the appropriate coordinates and bounds of integration to integrate the divergence. Calculate div(F) for F = <$ax3, 0, $az3>.

div(F) = U2GIbglD1oM=x2 + U2GIbglD1oM=z2

Substitute the expression for div(F) into the equation from the Divergence Theorem.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Since we need to compute the triple integral over the first octant of the sphere x2 + y2 + z2 = $c, we convert to spherical coordinates.

We convert using the standard spherical conversions.

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

Restricting the sphere to the first octant places the following bounds on ρ, θ, and φ.

0 ≤ ρ ≤

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

0 ≤ θ ≤

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

0 ≤ φ ≤

cxxSsJXaL0mNFeHkWEqCmG7R3LhAp6DWBpnocZl2JnKkDPdnNc43Az6kRfsZqysCzm/gj1yuLnVPi7BE976ekgGUasRD4bSmb9hMRggO6OPG3ojQBWL6OHjpijp/oRWeamYpfS5u9vFF81ez7DVRmHZN+iYYHtn4usdnBHDkETfZQ67KP6ATR0P0ekxifiYsregg2dcGGE/6e/iwbvo5b39zNs9caglp6BpkP6FoWns6TA53y599mCpk8RcjvAf076f/0L37829Bv49vSKOaQxsve+L2Uu/vb4UDtxdhRoms6D5lt5VwI+AemVg5nGJL0x/DyA5yr34jx92845H7uSZiKhK/qC1UdhHzj+aNKgT780GK4IzFRMTI8FKkBIUJWTYJqRltjE0fTMypXlaKrqXke0j9bfQBygx/DNcIN8GGaynodPzkkNsmLRBYhLioxZ60Ba++8dqfuglFkyVsDOVtwstSVS2ASjoGV0GlRz6zhX+5h6IqT7TUZdw71u4UKCsGLg==

Convert the integrand to spherical coordinates in terms of sin θ and sin φ. Recall the Pythagorean identity sin2x + cos2x = 1.

$dx2 + $dz2 = $d(ρ cos θ sin φ)2 + $d(ρ cos φ)2

= $dρ2(cos2θ sin2φ + cos2φ)

= $dρ2(cos2θ sin2φ + (0VV1JcqyBrI= − sin2φ))

= $dρ2(−sin2φ(0VV1JcqyBrI= - cos2φ) + 0VV1JcqyBrI=)

= $dρ2(0VV1JcqyBrI= − sin2φ sin2θ)

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Use information from Part 2 to write the appropriate triple integral. Recall that for spherical coordinates, the differential dV becomes ρ2 sin φ dρ dφ dθ.

=

/rLe4WZ0V+A6+5cdyAqcU1WzXLb5LhuxlqkZBje8SsyKwByEIQR4/g3uYPGhjBky3OvqicEHW/ePbdc32yFpwDfTmGPune1zMTYYBMHfOs/nz6I63I9ErOjLTsohKHZo71E9K0/r2T5qO7AO+ixUm1JM8S25zSMDHRcz++icgaagQ88LdlOdqn7JojdAMFh76HULB87LZvfmZcnDP81qPd/q4VwxpCVJ2SdRlyliiF0PjYjhIMnheahrhyk/VFEKQZHRIRkagx3zMTrM8A9aZt8BhcmvSbXUNZgu9Zdu1udd76tojSx7IFlTe6LVEYJphAV6+0K7rVz31nbxOvArnBhuIt2bkDNOv3UNhCmniqq4xvgTYrGYgz36+ReEnDLeJfL6i1kcRAC6BwQeNYklRyQ02NPou/77fJANQGA5UfhO0rQ7PGeITPyycVgejADxCRpO+6UPubDjcHS3arwvmBo60XLMIzM2/YkmULyz2ASpRR2CEitJLkDuIaOK+viAeIgw/rjEqZOz6W2XDRyeZfDuXJOckAQ4Nl9HmA2FNLZ+WtOBsJtNBZG5BwcIH8ntoYFikPnxgTY0dO9V2TP/NJK7RhUkipQlh+muFxJVrZcaS5kugkNl80FA028B5m9DehNDXqnFYhOeWRoW9EhF7cnxEMOXJJ3TZKCZY2WB/x6PTenSi6k8Q9hPRmBdovkJyBKAGeWx4iBJAPZQfhqbWx/KemxBSDTL47u3+AkcpPQRE1YI22W64m9TpkMfogagYtyV764c6TPjd4d4bqDq3qRRTy/MuRpHAsOQ1l3XzRc6MlMJAGxF6Kp0sLdWLFtWwIpUORSb5OoWVZw6GwB1nmrRYwMHMjCx7cyEWbg2ghMgJDqiEsdM/jubka/tI+4Ugbqpj6r2woPKHtpIQpYaOBgWhzgZTFwRq6zkwdf9n26ZGmPdiT1Y2kjwAj1PhzLReYfJgxPwiJJpeA58G38BvQCAp1TFyLz4zOl/P6f3E5Os3DrpSwFmirf0Lp02oA9X8a+rlfaRJiTJlI2WRi0bZmr1m4eWOHbtE9nxB5IZOdkY9Yu5zUxLpEEKDgRHnknD7oIRUgXOQTbe2raR7C7c5AxQD5w91VBkay0GDbNz2Itnx7MUMdTEPvilAKe5sQ3bZ6alhKgCUwU++j1BsPb6+ahF3U1oqODYkfAXtykNiD3mLH09pG9C4nXZvN0BpWfmkQaAA4jHaLzsyxhbmqrXtMyYKGcgRsFzP2h8oTt/DoQs22atMYidDUj0OeRUKTBG7AqWIkcb6zL+sdviqEAsSgc3zf/lhvClwd7htr2yWVaLo8pjMYjeLe4hoRoU8E940FWuQai49PQlGEQodgDx1CE4wlVTgVo40j4+RWaDPYWUlXDl9uMY/PdFlaW1rAZQp/vIGt9bBSzJALPTalXjJkgPeBQJciZzsY2NUNfsZEsELhZbk4245bqrZHEC+t7A+la2EUlXNkliFxozmjg5uxfwFkBfKAFljWsQpX/WwIT/21A9JdbSl1yoo+j9+VmM+Y6q1N8Xihk2+Ds/4ZnQJz1eeZ2AYlJEYhD/+s8tNeVM2oqfbGksvxKAe2KilOUlCrkqdv2tsw+NnRhvWx776zhjoVhoKDZj8mf8YuYYEELLE74e1rsaXPzwxd4ClHGRTtaqLfkTdQ1kFIAtm4qglsXovY18MZNPLWgx9oIsLgkd6OgNjesIIn+DErtzoK0t9i6ZjrUUTnnl7lzhVeEtMCX4IKyRMqeqFPQJIX1AlhFRElVPwiQTEs/6gpHDHOOdjao4EJqCdGPBh5wlIW3zm6PhQw6cXroGG1tnQWnfNkzXDkR9SmwccQDb7x1eUvcNwOlT0/fKzdzOzT7sCmDXz1y8Eo8Wo4nNu4heSQ==
Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Evaluate the flux by computing the triple integral.

= or6dmYWrEbA=π

Correct.
Incorrect.