I'm trying to check if $R/r$ is conservative feild
where $R=-x\hat i+y\hat j$ and $r=\sqrt{x^2+y^2}$
Attempt:
$$R/r=\underbrace{\frac{-x}{\sqrt{x^2+y^2}}}_{M}i+\underbrace{\frac{y}{\sqrt{x^2+y^2}}}_{N}j$$
$$\frac{\partial M}{\partial y}=\frac{xy}{\sqrt{x^2+y^2}(x^2+y^2)}$$
$$\frac{\partial N}{\partial x}=\frac{\color{red}-xy}{\sqrt{x^2+y^2}(x^2+y^2)}$$
Since $$\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$$
$R/r$ is not conservative field I think that I'm wrong
You are correct. There's a simple lemma which states that if the (scalar) curl of a vector field is non-zero, then the field is non-conservative.
Just as an FYI, some people mistakingly believe that the inverse of this statement is also true. But as it turns out a vector field being curl-free is not enough to say that it is conservative. However with the additional constraint that the domain of the vector field is simply connected, then we could say that such a vector field is conservative.