So there is a question that I did where it made me find the integral of $f \cdot dr$ under the path $C$ (which is a semicircle when $x=0$ so $y^2 + z^2 = $) for the arc that connects $(0,0,-1)$ to $(0,0,1)$. For clarity, the function $f$ is $(x^2 + y^2 + z^2 , -z, y+1)$. I evaluated this to be $\pi + 2$. Now, it asks whether $f$ is conservative, and the answer says : "No, the line integral is dependent on the path". Now I know this is exactly the definition of a conservative field (that it is independent of its path), but how can we come to this conclusion?
I evaluated it for the path that forms the full circle which is non-zero, so it mustn't be conservative, but is there another way? How do we show it is path dependent?
hint
It is conservative if
$$f_xdx+f_ydy+f_zdz=dU $$
or $$(x^2+y^2+z^2)dx-zdy+(y+1)dz=dU $$
then
$$U=-yz+C_1(x,z) $$ and $$U=yz+z+C_2(x,y) $$ which impossible.