Activity13.10.3.Checking the Visualization for Flux.
(a)
Figure 13.10.12 shows a plot of the vector field \(\vF=\langle{y,z,2+\sin(x)}\rangle\) and a right circular cylinder of radius \(2\) and height \(3\) (with open top and bottom). Consider the vector field going into the cylinder (toward the \(z\)-axis) as corresponding to positive flux.
Reasoning graphically, do you think the flux of \(\vF\) throught the cylinder will be positive, negative, or zero? Write a few sentences jutifying your answer.
Parameterize the right circular cylinder of radius \(2\text{,}\) centered on the \(z\)-axis, for \(0\leq z \leq 3\text{.}\) Be sure to give bounds on your parameters.
Based on your parameterization, compute \(\vr_s\text{,}\)\(\vr_t\text{,}\) and \(\vr_s \times \vr_t\text{.}\) Confirm that these vectors are either orthogonal or tangent to the right circular cylinder. Is your orthogonal vector pointing in the direction of positive flux or negative flux?
Write a couple of sentences to explalin how the results of the flux calculations would be different if we used the vector field \(\vF=\left\langle{y,z,\cos(xy)+\frac{9}{z^2+6.2}}\right\rangle\) and the same right circular cylinder.
write a couple of sentences to explain how the results of the flux calculations would be different if we used the vector field \(\vF=\langle{y,-x,3}\rangle\) and the same right circular cylinder.
