We give ourselves a choice of integrals to compute.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

For each of the following problems, use either the divergence theorem or Stokes’ theorem to set-up integrals that will compute the desired result.

Compute the flux of
\[ \vec {F}(x,y,z) = \vector {x,-y,z} \]
across the unit cube in the first octant.
Compute the circulation of
\[ \vec {F}(x,y,z) = \vector {3x,4y,5z} \]
along the surface defined by the lower hemisphere of \(x^2 + y^2 + z^2 = 4\).
Compute the flux of
\[ \vec {F}(x,y,z) = \vector {-y,x-z,y} \]
across the cone of radius \(6\) and height \(2\) whose base is in the \((x,y)\)-plane.
Compute the circulation of
\[ \vec {F}(x,y,z) = \vector {x+y,y+z,x+z} \]
along the surface enclosed by the tilted circle
\[ \vec {p}(t) = \vector {\cos (t), 2\sin (t),\sqrt {3}\cos (t)}. \]