Question

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = <<z - x, x - y, 2y - z>>, D is the region between the spheres of radius 2 and 4 centered at the origin.

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D.
\(\displaystyle{F}={\left\langle{z}-{x},{x}-{y},{2}{y}-{z}\right\rangle}\), D is the region between the spheres of radius 2 and 4 centered at the origin.

Answers (1)

2020-11-09
Step 1
Given vector field is \(\displaystyle{F}={\left\langle{z}-{x},{x}-{y},{2}{y}-{z}\right\rangle}\).
Compute the derivative of the above vector field.
\(\displaystyle\nabla\cdot{F}=\frac{\partial}{{\partial{x}}}{\left({z}-{x}\right)}+\frac{\partial}{{\partial{y}}}{\left({x}-{y}\right)}+\frac{\partial}{{\partial{z}}}{\left({2}{y}-{z}\right)}\)
=-1-1-1=-3
Step 2
Now calculate the net outward flux.
\(\displaystyle\int\int_{{D}}\int\nabla\cdot{F}{d}{V}=\int\int_{{D}}\int{\left(-{3}\right)}{d}{V}\)
=-3(Volume of a sphere)
\(\displaystyle=-{3}{\left(\frac{{4}}{{3}}\pi{\left({{r}_{{2}}^{{3}}}-{{r}_{{1}}^{{3}}}\right)}\right)}\)
\(\displaystyle=-{4}\pi{\left({4}^{{3}}-{2}^{{3}}\right)}{\left[\because{r}_{{1}}={2},{r}_{{2}}={4}\right]}\)
\(\displaystyle=-{224}\pi\)
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