Simplifiyng the proof of conservative field using rottor properties

[slonopotam]

$$\vec{F}=(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}})$$
$$\vec{r}=(x,y,z)$$

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

$$\vec{F}=(\frac{x}{|r|},\frac{y}{|r|},\frac{z}{|r|})$$

so its $$F=\frac{r}{|r|}$$

i need to prove that F is a conservative field
where (x,y,z) differs (0,0,0)
so i need to show that rot f is 0
but for rottor i need a determinant
is there a way to do a rot on simpler way?

 

[jpreed]

A conservative field is also irrotational. Take the curl of the field and show that it is equal to zero. (It is).