Exercise 18

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 $x\geq 0$.