# Why does the domain affect whether or not a vector field is conservative?

I'm wondering if there is some intuitive way of understanding why the specified domain has the "power" to make a vector-field conservative or not.

e.g. $\quad \displaystyle \boldsymbol{F}(x,y) = \frac{-y \, \boldsymbol{ i} + x \, \boldsymbol{ j}}{x^2 + y^2}$ is conservative on the plane minus a ray coming out of the origin (let's say the $x$-axis for specificity).

The fact that including / excluding such a ray would affect something that appear to be a property of the field itself is really counter-intuitive to me; especially when applied to physical fields like the gravitational or electromagnetic fields - aren't these conservative regardless of what points we include in space?

• As a further point to illustrate my confusion further: if we consider the above vector-valued function to be defined on $\mathbb{R}^3$, then wouldn't it be conservative? And yet wouldn't some of the possible paths overlap with paths that could be defined only in $\mathbb{R}^2$? I don't see how the two situations would differ, mathematically, and yet the definition seems to imply that the line integral of such a field would (could?) differ in these two situations... – Rax Adaam Feb 19 '16 at 1:16