How can I find this surface integral in cylindrical coordina

[John004]

Homework Statement

A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by

$\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda z)\hat k $

where $\hat i$ , $\hat j$ , and $\hat k$ are the unit vectors along the cartesian axes and $\hat \rho$ is the unit vector $\frac{x}{\rho}\hat i \ + \frac{y}{\rho}\hat j$.

(a) Calculate, as a surface integral, the flux of $\vec F$ through the closed surface bounded by the cylinders $\rho = a$ and $\rho = 2a$ and the planes $z =\pm \frac{a\pi}{2} $.

(b) Evaluate the same integral using the divergence theorem.

So I am pretty lost on how to even begin part (a). I sketched a picture of the situation and am I kinda confused as to how to represent the unit normal vector, as well as the infinitesimal area elements. Am I supposed to take four separate integrals and sum them afterwards, one for the inner and outer curved surfaces, and one for the upper and lower caps?

 

[LCKurtz]

I see that this is marked solved, so I haven't commented. I just fixed the text to make it readable. @John004: use ## instead of $ to delimit inline Latex.

Homework Statement

A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by

$$\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda z)\hat k $$

where $\hat i$ , $\hat j$ , and $\hat k$ are the unit vectors along the cartesian axes and $\hat \rho$ is the unit vector $\frac{x}{\rho}\hat i \ + \frac{y}{\rho}\hat j$.

(a) Calculate, as a surface integral, the flux of $\vec F$ through the closed surface bounded by the cylinders $\rho = a$ and $\rho = 2a$ and the planes $z =\pm \frac{a\pi}{2}$.

(b) Evaluate the same integral using the divergence theorem.

So I am pretty lost on how to even begin part (a). I sketched a picture of the situation and am I kinda confused as to how to represent the unit normal vector, as well as the infinitesimal area elements. Am I supposed to take four separate integrals and sum them afterwards, one for the inner and outer curved surfaces, and one for the upper and lower caps?