${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$where $S$is the portion of the hyperboloid of two sheets $x^2+y{}^2+1=z^2$that lies between the planes $z=5$and $z=10$and where

Consider the vector field below:

The goal is to determine whether the integral ${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$can be evaluated using the Divergence Theorem, with the surface $S$ being the section of the hyperboloid of two sheets $x^2+y{}^2+1=z^2$lying between the planes $z=5$and$z=10$.

Divergence According to the theorem,

"Let $W$be a bounded region in $R^3$with a smooth or piecewise-smooth closed oriented surface as its boundary $S$. If on an open region containing W, a vector field $F(x,y,z)$is defined, then

${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}dS={\iiint }_{W}div\mathbf{F}(x,y,z)dV."........\left(1\right)$

where $n$ is the outwards unit normal vector. "

The piecewise-smooth surface $S$is the section of the hyperboloid of two sheets $x^2+y{}^2+1=z^2$that falls between the planes $z=5$and $z=10$. Also defined in the region enclosed by $S$is the vector field

Divergence Only smooth or piecewise-smooth surfaces with a vector field $F(x,y,z)$defined on the region encompassed by $S$can be used to prove the theorem.

Yes, the integral ${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$, can be evaluated by means of Divergence Theorem.