the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$
first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I need to show that certain conditions of the parametrization. In other words, can I show this surface is smooth without finding the parametrization of it ?
second Q: how can I find the parametrization of this surface ? I know the parametrization of a sphere (http://www.wolframalpha.com/input/?i=parametrization+of++sphere ) , so I tried to use it in, for example we can deduce $|cos(v)| < 0.8$, but other condition seem complicated.. so I doubt this is the way to have it.
Answer to Q1) Nope. In order to use divergence theorem, the given area should be a boundary of some 3-dimensional object. Instead, you can use Stokes' theorem with $\partial S$ composed with 6 circles of radius 3. (The subset with $|x|,|y|,$ or $|z|=4$) $S$ is a smooth surface with boundary because in any point on the interior of $S$ you can find a local parametrization, for example the one restricting the parametrization of sphere.
Answer to Q2) I think there does not exist single parametrization with barely tolerable expression. What you mentioned is one possibility, or you can use parametrization $(x,y)\mapsto (x,y,\pm\sqrt{25-x^2-y^2}), |x|,|y|\leq 4$.