calculate surface integral of a cylinder with rectangular vector

I am lost with that problem, and I cannot continue.

The problem asks to calculate that:

$ $ \oint_S\ F.dS$ $ Being:

$ $ F = (x^2, y^2, (z^2-1))$ $

and S is defined by these Cylindrical coordenates: $ $ r = 2; 0<z<2; 0\leqΦ\leq2π$ $

I converted F to cylindrical coordenates to have both in the same system. I figured out for cylindrical system: $ $ dS = (r.dΦ.dz)\hat ar + (\hat aΦ + (r.dr.dΦ) \hat az$ $

But is that the best way to do that with surface integrals? Seems, that integral gives a lot of job. Plese help me.

Arithmetic that corresponds to combinatorial rectangles and cylinder intersections?

Definable subsets of $ \mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets.

In communication complexity the interpretation is more on intersection and union of combinatorial rectangles or complements of combinatorial cylinder intersections which does not seem to be as nice as what comes from geometric interpretation of Presburger.

Is there an arithmetic that corresponds to definable sets in communication complexity?

Would it be reasonable to expect something?