why is the curl of a function (which looks like a vortex) zero?

Question

Can anyone explain why the curl of this function zero ? Clearly it's rotating then how is it irrotational ? Someone on SE said

Not to mistake Curl for rotation, they aren't exactly similar.

Is this true ?

$\vec{V}(x,y,z)=[\frac{−y}{x^2+y^2},\frac{x}{x^2+y^2},0]$

Image Source: WolframAlpha

Edit : As pointed by Rahul the topic of irrotational vortices have been already discussed over here, that somehow clears my intuitive understanding of curl: rotating the particles about their axes and not just revolving around center (which Travis has already pointed out in his answer) of the vortex but this still doesn't clear up my doubt about why the intuitive understanding doesn't match the calculations.

If the curl is present at the center (z-axis), why doesn't it show up in the result (which is zero)?

Answer 1

It's true that the $\bf V$ is "rotating" in that sense that, informally, if you place a particle in $\Bbb R^3 - \{\textrm{$z$-axis}\}$ and let it move with velocity the value of $\bf V$ wherever it is, after time $t$ it will have rotated anticlockwise $t$ radians about the $z$-axis.

However, $\operatorname{curl}$ doesn't measure this sort of rotation---rather, it measures the infinitesimal rotation of the vector field about each point, and for the particular vector field $\bf V$ this rotation is zero about every point.

Answer 2

Curl measures the local rotation about a point - or better, it measures the extent to which the field locally fails to be conservative: that is, if we consider this a field of force, like a gravitational or electric field, and I put a charged particle, or particle that is otherwise responsive to this force, in it and then take it around a tiny loop (ideally: infinitely small), do I get any net work and, if so, how much? If the curl vanishes everywhere on a simply-connected domain, the field is conservative on that domain.

It is thus a complicated kind of differential operator - an operator that reveals a local property of a function: a property that holds within an ideally small ("infinitesimal") area around a point. The simplest one is the derivative in one variable, which reveals how a function stretches/squeezes about that point The existence of a differential operator at a point implies a certain regularity, while its non-existence implies a certain pathology in that area. Thus it should not be surprising that $\mathbb{curl}\ \mathbf{F}$ can be written as $\nabla \times \mathbf{F}$.

In the case of the origin point, the curl is undefined and thus it tells you nothing about whether or not any work can be done in moving thereabout. The point is pathological as the function is not defined there and moreover the local neighborhood looks dramatically different. In this case, it turns out you can't do any work, but there are other cases where you can (even while the curl is zero elsewhere, meaning that a loop around them that is small enough to exclude the origin point will not achieve any work). The domain of this field is not simply connected as the origin is missing.