Can anyone suggest an example of a vector field $F: A \subset \mathbb{R}^3 \to \mathbb{R}^3$ that satisfies all the following conditions?
(Also an example in dimensions other than $\mathbb{R}^3$ would be good)
On
The field $F$ is conservative, so it has antiderivative: we have $F=\mathrm{grad}\,\varphi$ with some scalar field $\varphi$. If $F$ is differentiable at some point $a$, then $\mathrm{curl}\,F(a)=0$ by Young's theorem. Therefore, curl is either zero or undefined.
An example for undefined curl is: $$ F(x,y,z,\ldots) = \big(|x|,0,0,\ldots\big). $$ This field has the antiderivative $\frac{x|x|}{2}$ so it is conservative; but $\mathrm{curl}\,F$ is undefined along the plane $x=0$.
If you want to define curl using only partial derivatives (although curl should be independent from the co-ordinate system), a modified construction is $$ F = \mathrm{grad} \frac{(x+y)|x+y|}2 = \big( |x+y|,|x+y|, 0 \big). $$
Here we give an example in $\mathbb{R}^2$. Generalizing to $\mathbb{R}^3$ is straightforward. Let $$\varphi = \begin{cases} \frac{x y(x^2-y^2)}{x^2+y^2}, & (x,y)\ne (0,0) \\ 0, & \textrm{else} \end{cases}$$ and consider the gradient field ${\bf F}=\nabla\varphi$ on $\mathbb{R}^2$. It is a standard exercise to show that $\mathrm{curl}\,{\bf F}$ is undefined at the origin.