Show that $$\vec F=\left(\frac{y + z}{x^2+y^2+z^2}, \frac{x + z}{x^2+y^2+z^2}, \frac{y + x}{x^2+y^2+z^2}\right)$$ in $\mathbb R^3-\{(0,0,0)\}$ is irrotational but not conservative.
I can show that $\vec F$ is irrotational by proving $curl\,\vec F=0$.
Please help me to show that $\vec F$ is not conservative.
The line integral for $C:\vec r=(acos t, a sint, 0)$, $0\leq t\leq 2\pi$, circle centered at origin is $$\int F.dr=\int_0^{2\pi}cos\, 2t\,dt=0$$
We have to consider a closed curve such that $\int F.dr\neq 0$, then we are done!!!
But how to show this?
Hint: $\vec{F}$ is not irrotational: \begin{align} \nabla\times\vec{F}=\begin{pmatrix}\partial_yF_z-\partial_zF_y\\\partial_zF_x-\partial_xF_z\\\partial_xF_y-\partial_yF_x \end{pmatrix} \end{align} and \begin{align}\require{cancel} \partial_yF_z-\partial_zF_y&=\frac{\cancel{x^2+y^2+z^2}-(y+x)2y-\cancel{(x^2+y^2+z^2)}+(x+z)2z}{(x^2+y^2+z^2)^2}\\ &=2\frac{-y^2-xy+xz+z^2}{(x^2+y^2+z^2)^2}\,,\\[2mm] \partial_zF_x-\partial_xF_z&=2\frac{-z^2-yz+xy+x^2}{(x^2+y^2+z^2)^2}\,,\\[2mm] \partial_xF_y-\partial_yF_x&=2\frac{-x^2-xz+yz+y^2}{(x^2+y^2+z^2)^2}\,. \end{align}
To address the comment that you are searching for a field in $\mathbb R^3$ that is irrotational but not conservative.
As Travis Willse pointed out: when the only singularities are isolated points every irrotational field in $\mathbb R^3$ is conservative.
What you have to look for is a field that has a singularity on a line such that it is not possible to contract a loop around that line to a single point. Take for example the $z$-axis. The most popular irrotational field that is not conservative is $$ \vec{F}=\frac{1}{x^2+y^2}\begin{pmatrix}-y\\x\\0\end{pmatrix}\,. $$