Simple integral in cylindrical coordinates

[Vagrant]

Homework Statement

As a part of bigger HW problem, I need to calculate the integral:
$\oint$[$\hat{r}$+$\hat{z}$]d$\phi$

Homework Equations

The Attempt at a Solution

In cylindrical coordinates:
=[$\hat{r}$+$\hat{z}$] $\oint$d$\phi$
=2∏[$\hat{r}$+$\hat{z}$]

On the other hand if I convert it to Cartesian coordinates:
$\oint$[cos$\phi$$\hat{x}$+sin$\phi$$\hat{y}$+ $\hat{z}$]d$\phi$
=2∏$\hat{z}$

So, what is it that I am doing wrong in the case with cylindrical coordinates? I'm sure I'm missing something very basic.
Thanks.

 

[SammyS]

Homework Statement

As a part of bigger HW problem, I need to calculate the integral:
$\oint$[$\hat{r}$+$\hat{z}$]d$\phi$

Homework Equations

The Attempt at a Solution

In cylindrical coordinates:
=[$\hat{r}$+$\hat{z}$] $\oint$d$\phi$
=2∏[$\hat{r}$+$\hat{z}$]

On the other hand if I convert it to Cartesian coordinates:
$\oint$[cos$\phi$$\hat{x}$+sin$\phi$$\hat{y}$+ $\hat{z}$]d$\phi$
=2∏$\hat{z}$

So, what is it that I am doing wrong in the case with cylindrical coordinates? I'm sure I'm missing something very basic.
Thanks.

$\hat{r}$ is a function of $\phi\ .$

 

[Vagrant]

Sorry, but could you please explain this a bit?
I can visualize that $\hat{r}$ is a function of $\phi$(unit circle on x-y plane centered at origin) for cartesian coordinates, but I'm drawing a blank when it comes to cylindrical coordinates.