Divergence theorem...(multivariable calculus)

S is the surface of the solid bounded by the cylinder x^2+y^2= 1 and planes z=0 and z=2.

use the divergence theorem to calculate the surface integral where vector field F= .


My thoughts so far:

So I got my divF = 3x^2 + 3y^2+ 3z^2

then for my triple integral i said the bounds are {r,t,z: 0
Is this correct? if so, do i need to add the "r" in (r)(dz)(dr)(dt)

And in general how do I know when to add the extra r if this even applies? I am not exactly sure if this is what is called a Jacobian, but I am confused in when to put this.

Thanks for the help!

You are correct.

If you use polar or cylindrical coordinates, the jacobian equals r; you need to always include this factor in these conversions.

Jacobian for cylindrical coordinates:
Since x = r cos θ, y = r sin θ, z = z, the jacobian ∂(x,y,z)/∂(r, θ, z) equals
|..cos θ.....sin θ....0|
|-r sin θ...r cos θ...0| = r.
|.....0..........0.......1|

I hope this helps!