$(1$ point$)$ This problem will illustrate the divergence theorem by computing the outward flux of the vector field $F(x,y,z)=2+2yj+2zk$ across the boundary of the right rectangular prism: $-2x3,-2y4,-3z4$ oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism.

Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism to be positive.

Part $1-$ Using a Surface Integral

First we parameterize the six faces using $0s1$ and $0t1$ :

The face with $z=-3$:${}_1=(x_1(s),y_1(t),z_1(s,t))$

$x_1(s)=$

$y_1(t)=$

$z_1(s,t)=-3$

$$

$$

The face with $z=4$:${}_2=(x_2(s),y_2(t),z_2(s,t))$

$x_2(s)=$

$y_2(t)=$

$z_2(s,t)=4$

$$

$$