The Divergence Theorem

1. Find the flux of the vector field $\vF = \langle 3x^2+y^5,5+e^{z^3},z\rangle$ through the surface of the solid cube $Q$ in $\R^3$ with $-2\leq x\leq 2$, $-2\leq y\leq 2$, and $-2\leq z\leq 2$.

2. Find the flux of the vector field $\vG = \langle x^3,y^3,z^3\rangle$ through surface consisting of the top half of sphere of radius $3$ centered at the origin and the disc of radius $3$ in the $xy$-plane (centered at the origin).

Let $Q$ be the volume enclosed by $x=0$, $x=1$, $y=0$, $y=1$, $z=0$, and $z=1$. Compute the flux of $\vF = \langle z^2-xy,4yz+cos(x/\pi),e^{xyz} \rangle$ through each of the six cube faces.

Let $Q$ be the volume given in cylindrical coordinates by $0 \leq z \leq 3$, $1 \leq r \leq 2$, and $0 \leq \theta \lt 2 \pi$. Give an example of a vector field with component functions that are linear in $x$, $y$, and $z$ such that the flux of the vector field through the boundary of $Q$ is 17.

Let $Q$ be the volume given in spherical coordinates by $0 \leq \rho \leq 3$, $0\leq \phi \leq \pi/4$, and $0\leq \theta \lt 2 \pi$. Give an example of a vector field with component functions that are not linear or constant such that the flux of the vector field through the boundary of $Q$ is 25.

Calculate the flux of the given vector field $\vF$ through the surface $S$ for each situation below by using an appropriate triple integral from the Divergence Theorem.

1. The vector field $\vF = \langle 2x+3\sin(yz),-4y+e^{x^2} , 7z+\arctan(y^5)$ through the tetrahedron $S$ with vertices $(0,0,0)$, $(1,0,0)$, $(0,2,0)$, and $(0,0,2)$.

2. The vector field $\vF=\langle xy^2,yz^2,zx^2\rangle$ through $S$, the portion of the sphere of radius $3$ centered at the origin having $x\geq 0$.