Ok I've come across a similar question which I am having troubles with, this time its the method

A vector field exists so that:

$\displaystyle \vec{F(r)}=f(r)\vec{r}$

where

$\displaystyle r = \sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

C is the path on the sphere which is $\displaystyle {x}^{2}+{y}^{2}+{z}^{2}=1$

Prove that $\displaystyle \int \vec{F}.d\vec{r} = 0$

Now im not entirely sure how to best approach this, i can easily see:

$\displaystyle \int \vec{F}.d\vec{r} = \int f(r)\vec{r}.d\vec{r}$

Though i am unsure how to encorporate the path to deduce zero.

I guess the main thing throwing me here is there f(r) i have r, and i could use sphereical coordinates, but I do not know f(r), just part of it (the r). Do i assume f(r) = r?? Then convert into sphereical coordinates, work out normal etc ?