Flux Integral Over Cylinder

IniquiTrance
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?

~Death~
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.

IniquiTrance
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.