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:

$$\oint_S\ \vec F\cdot \vec {dS}$$ Being a vector defined by:

$$F = x^2\hat a_x+ y^2\hat a_y + (z^2-1)\hat a_z$$

and S is defined by these Cylindrical coordenates: $$r = 2; 0

I converted F to cylindrical coordenates for having both in the same system. I found for cylindrical system, derivative area is given by: $$dS = (r.dΦ.dz)\hat ar + (dr.dz)\hat aΦ + (r.dr.dΦ) \hat az$$

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

• What exactly are these expressions $\hat ar, \hat a\Phi, \hat az$ at the end? (By the way - you should go check the wording of your problem carefully to determine if $S$ is eactly what you've discussed here, or should it also include the dlsks at $z = 0$ and $z=2$ that would close off the surface. If so, then you have two more integrals to do.) – Paul Sinclair Mar 30 at 0:49
• And a hint: Note that $r$ is constant on $S$. – Paul Sinclair Mar 30 at 0:51
• These expression $\hat ar,\hat aΦ,\hat az$ are derivation of the surface area of a cylinder, written in Cylindrical Coordinates. But I do not need to use in $Φ$ direction, once I need to calculate area of top, bottom and side. – user2793412 Mar 30 at 3:54
• Are you familiar with differential forms? – Bertrand Wittgenstein's Ghost Mar 30 at 4:52
• That doesn't answer my question. Note the "exactly" part. How are you defining these? They look like you are multiplying the same vector by 3 coordinates, which makes no sense at all. – Paul Sinclair Mar 30 at 14:14

By divergence theorem, we can write$$\oint_S \vec F\cdot \vec{dS}=\iiint_V \nabla\cdot fdV$$where$$\nabla\cdot f{=2x+2y+2z\\=2r(\sin\phi+\cos \phi)+2z}$$