Brain Gains (Rapid Recall, Jivin' Generation)

  • Draw the surface $S$ with parametrization $\vec r(u,v) = (u, u\cos v, u\sin v)$ for $1\leq u\leq 3$ and $0\leq v\leq \pi$.
  • Set up the surface integral that would give the surface area of $S$.
  • Set up the surface integral formula that would give the $z$-coordinate of the centroid of the surface $S$.
  • Set up the surface integral that gives the flux $\Phi = \iint_S \vec F\cdot \hat n dS$ of the vector field $\vec F(x,y,z) = (2x+y,-yz^2, 3z)$ across the surface $S$ in the direction that points away from the $x$-axis.

Solutions

The solutions are all coded using the Mathematica code below. Note that the vector $n$ has a positive $x$-component, which means that the vector points inward towards the $y$-axis. The negative sign in the code below is needed to correctly get the flux. 

r = {u, u Cos[v], u Sin[v]}
ParametricPlot3D[r, {u, 1, 3}, {v, 0, Pi}]
n = Cross[D[r, u], D[r, v]] // Simplify
Integrate[Norm[n], {u, 1, 3}, {v, 0, Pi}]
Integrate[u Sin[v] Norm[n], {u, 1, 3}, {v, 0, Pi}]/Integrate[Norm[n], {u, 1, 3}, {v, 0, Pi}]
Integrate[{2 u + u Cos[v], -u Cos[v] (u Sin[v])^2, 3 u Sin[v]} . -n, {u, 1, 3}, {v, 0, Pi}]
Integrate[ReplaceAll[{2 x + y, -y z^2, 3 z} . -n, {x -> u, y -> u Cos[v], z -> u Sin[v]}], {u, 1, 3}, {v, 0, Pi} ]

Show[
 ParametricPlot3D[r, {u, 1, 3}, {v, 0, Pi}],
 VectorPlot3D[{2 x + y, -y z^2, 3 z}, {x, 1, 3}, {y, -3, 3}, {z, 0, 3}]
 ]

Group Problems

  1. Consider the surface $S$ parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$.
    1. Draw the surface.
    2. Compute $dS = \left|\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
    3. Set up an integral formula to compute the surface area of $S$.
    4. Set up an integral formula to compute $\bar z$ for this surface.
    5. We would like an orientation $\hat n$ for the surface that points away from the $z$-axis. Does $ \dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}$ point towards the $z$-axis, or away from the $z$-axis?
    6. Set up the surface integral that gives the flux of $\vec F = (3yz,-2x+y, z-2x)$ across the surface $S$ in the direction of $\hat n$. Then use software to compute the integral.
  2. Consider the parametric surface $\vec r(u,v) = (u, u\cos v,u\sin v)$ for $0\leq v\leq \pi$ and $0\leq u\leq 4$.
    1. Draw the surface.
    2. Compute $dS = \left|\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial v}\right|dudv$.
    3. Set up an integral formula to compute $\bar x$ for this surface.
    4. Give an equation of the tangent plane to the surface at $(u,v) = (1,\pi/2)$.