As an example, in order to apply the Gauss's theorem, one must have a closed surface. Hence, if one is given the equation of an open surface S: $$1-z=x^2 + y^2 , 0\le z \le1$$ and if we want to calculate, using Gauss's theorem $$\int_S \bar F\cdot d\bar S$$ then we will need to construct a closed surface including the surface S given above. I realise that this is a circle on the xy plane and I am able to solve this problem.
However, my question is, more generally, are there any standard procedures or method which allows one to construct a closed surface from an arbitrary equation of an open surface, without the need to first visualise the surface given? An example of such a method in the answer will be much appreciated.
Thank you for your help.