Task 47.1

From Ben: This task is now ready to work on.

Consider the vector field $\vec F = \frac{ (x,y,z) }{ (x^2+y^2+z^2) ^{3/2}}$. This vector field is directly proportional to gravitational and electric fields. In this task, we will show that flux of this field across any closed surface that contains the origin is $4\pi$, while the flux across any surface that does not contain the origin is zero.

  1. Show that the divergence of $\vec F$ is 0, provided $(x,y,z)\neq(0,0,0)$. You can to this by hand, or with software, or find the solution somewhere online (one of the links in 44.1 has the work).
  2. Consider the surface $S$ which is a sphere of radius $a$ (so $x^2+y^2+z^2=a^2$). Compute the outward flux of $\vec F$ across $S$, and show that you get $$\Phi = \iint_S \vec F\cdot \hat n dS = 4\pi.$$ In particular, notice that the answer does not depend on the radius of the sphere. There are multiple ways to do this. One option is to obtain a parametrization of the surface (look back at previous problems to get a parametrization) and then set up and compute the integral directly. Another is to reason out geometrically what $\hat n$ must equal because we're on a sphere of radius $a$, and then replace $x^2+y^2+z^2$ with $a^2$ in many places.
  3. Let $S$ be any closed surface which does not contain $(0,0,0)$ on the surface, or inside the surface. Explain why $\iint_S \vec F\cdot \hat n dS =0$
  4. Let $S$ be any closed surface which does contain $(0,0,0)$ inside the surface. Explain why $\iint_S \vec F\cdot \hat n dS =4\pi$. (Start by picking a small radius $a$ so that the sphere $x^2+y^2+z^2=a^2$ lies entirely inside $S$.)

Task 47.2

From Ben: This task is now ready to work on.

Let $D$ be the solid region in space that lies above the cone $z^2=x^2+y^2$ and below the paraboloid $z=6-x^2-y^2$. In cylindrical coordinates, the domain $D$ lies above the cone $z=r$ and below the paraboloid $z=6-r^2$. Let $\vec F = (x^2, 4y+z, z-x+3)$. Verify the divergence theorem for this solid region. A parametrization for the cone is $\vec r(u,v) = (u\cos v, u\sin v, u)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$. A parametrization for the parabaloid is $\vec r(u,v) = (u\cos v, u\sin v, 6-u^2)$ for $0\leq u\leq 2$ and $0\leq v\leq 2\pi$.

  • Compute $\iint_S \vec F\cdot \hat n dS$ across the cone, making sure you have an outward pointing normal vector.
  • Compute $\iint_S \vec F\cdot \hat n dS$ across the paraboloid, making sure you have an outward pointing normal vector.
  • Compute $\iiint_D \vec \nabla \cdot \vec F dV$, and show how to combine the results of the previous computations to get the same value.

In my work above, I ended up seeing the values $-20\pi/3$, $60\pi$, and $160\pi/3$.

Task 47.3

Pick a problem from OpenStax from 6.7 or 6.8, where it asks you to verify either Stokes' or the Divergence theorem, and complete it.

Task 47.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.