"Determine by appropriate calculations whether a force $\vec{F}$ acting on a particle at $\vec{r}$ from the source by the relation: $$ \vec{F}(\vec{r}) = \frac{A}{r^2} \frac{\vec{r}}{|\vec{r}|} $$ (A is a constant) is conservative or not."
I tried splitting the function into $x, y, z$ components with no success. I was hinted that I should solve it by trying a circular path and seeing if the work is indeed $0$, however I dont know how to approach it. Any help is much appreciated.
On
This is most easily done in spherical co-ordinates.
see http://hyperphysics.phy-astr.gsu.edu/hbase/gradi.html#c2
for the appropriate formulas.
In your case ... $$\vec{F}(\vec{r}) = \vec \nabla f(\vec r)$$
where $f(r)$ is the scalar function $$ f(\vec r)=-\frac A r $$
So $ \vec{F}(\vec{r})$ is conservative because $$ \int _{\vec a} ^{\vec b} \vec{F}(\vec{r})\cdot d\vec r = f(\vec b) - f(\vec a)$$
independent of path.
You can decompose the path into infinitesimal segments in radial direction and infinitesimal arcs, perpendicular to the radius. Along those arcs $d\vec r$ is perpendicular to $\vec r$, so the work done is $dW=\vec F(\vec r)\cdot d\vec r=0$. Therefore the integral along any circular segment centered on origin will be $0$. So all you need to worry is the integral in the radial direction for those segments $d\vec r$ is along $\vec r$, so $dW=\frac A{r^2} dr$. When you integrate this, you will get $0$.