Let $S$ be the hemisphere $x^2+y^2+z^2=4,z0.$ Compute

${\displaystyle \underset{S}{\overset{\phantom{}}{}}}{z}^{2}d$

Let $D$ be the sphere centered at the origin with radius $1.$ Compute the flux of the vector field

$F(x,y,z)=($:$x^2,xz,3z$:$)$

through the boundary of this sphere. Hint: use the divergence theorem.

Let $C$ be the circle $x^2+y^2=4,z=3,$ oriented counter$-$clockwise, and let

$G(z,y,z)=($:$2y,3x,-z^2$:$).$

Compute

$o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}G*dr.$

Hint: Use Stokes' Theorem.