${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$

where $S$is the unit sphere and$\mathbf{F}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}<e\sqrt{x^2+y^2+z^2},\ln \left(x^2{y}^2{z}^2\right),5x^{3}z>$

Consider the vector field below:

The goal is to determine whether the integral ${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$, where the surface $S$ is the unit sphere, can be evaluated using the Divergence Theorem.

Divergence According to the theorem,

"Let $W$be a bounded region in role="math" localid="1650773931079" $R^3$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 $n$ denotes the outwardly unit normal vector

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.

The unit sphere's surface $S$is a smooth surface.

The vector field, however, is not defined across the region enclosed by $S$.

No, the integral ${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}ds$, cannot be evaluated by means of Divergence Theorem.