Flux Integral Over Cylinder

To compute the flux integral over a cylinder's walls oriented along the z axis:

Can I do:

$$\int\int \vec{F}\cdot\nabla G(x,y,z) dA$$

$$G(x,y,z) = r^{2}=x^{2}+y^{2}$$

$$\nabla G = <2x, 2y, 0>$$

$$\int\int \vec{F}\cdot <2x,2y,0> dA$$

Is this a correct approach?

To compute the flux integral over a cylinder's walls oriented along the z axis:

Can I do:

$$\int\int \vec{F}\cdot\nabla G(x,y,z) dA$$

$$G(x,y,z) = r^{2}=x^{2}+y^{2}$$

$$\nabla G = <2x, 2y, 0>$$

$$\int\int \vec{F}\cdot <2x,2y,0> dA$$

Is this a correct approach?
Assuming a positive orientation, the easiest way to do it is by Divergence Theorem.

(1) Find the divergence of $$\vec{F}$$

(2) Integrate this over the solid cylinder.

The other way is to split the cylinder into 3 pieces the Top, Bottom and Side and the sum the flux contributed from each piece.

Assuming a positive orientation, the easiest way to do it is by Divergence Theorem.

(1) Find the divergence of $$\vec{F}$$

(2) Integrate this over the solid cylinder.

The other way is to split the cylinder into 3 pieces the Top, Bottom and Side and the sum the flux contributed from each piece.
Thanks for the reply.

Yeah, I specifically want to solve it as a flux integral without the div theorem.

Also know how to split it up. Is this a proper way to compute it over the cylinder walls though?