Circulation of Vector Field versus Work done by Vector Field According to wikipedia, the circulation of a vector field along a curve is defined as:
$$\oint V \cdot dl$$
where $V$ is the vector field, and $dl$ is the infinitesimal component along the  boundary of the curve.
Isn't this analogous to work? Where work is defined as:
$$ \oint F \cdot dr $$
Where $F$ is the vector force field and $dr$ is the infinitesimal path component? And if so, why is there a distinction in the naming?
 A: Question makes sense. In the most general case, it is analogous to work only it in that it is a continuous sum of the infinitesimal direction component of displacement across another vector. In some specific cases they are very similar, and in some cases that are the same exact thing.
A circulation is a path integral in a vector field around a closed curve. Work is a path integral of force around a curve. Just those sentences themselves tell what is different - first sentence has “vector field” and “closed”, whereas second sentence does not. Second sentence has “force” and first doesn’t.
So those three differences, which do not always apply:
1.In a vector field, $V$ is unchanging. No matter how we move about in that field, the $V$ magnitude and direction will be only a function of position. That’s what a vector field is.
This does not have to be true about work. The path taken may be part of what determines the force. Or there may be a time component. The vector may change through time. In a vector field it’s unaffected by time, path, everything except position.
In some cases, forces are a vector field. A spring, or in an electric or magnetic field, or slowly in a fluid pressure field. Many situations.
2.In a circulation, the vector field isn’t always force. It is as often as not something else. Electric field for example. But your question assumes and implies this already, as you are only asking if it is analogous. If it isn’t force, then the integral of the vector dot displacement has a different meaning, but is otherwise a similar thing (how youve moved in direction of the vector and its magnitude).
3.If you had asked about a path integral in a vector field and how that might be analogous to work, then the question would be simpler, because we could stop there. But the link goes to a circulation, which is a path integral around a closed loop. So the circulation only applies to a loop where you end-up where you started. There are situations where work is calculated over a closed loop in a vector field, and that would be exactly “a circulation”, but they are less common.
If that was the question, then we’d be comparing “a path integral in a vector field”, and “a path integral over force”. So if we were moving around in a force field that was characterized by location, like the ones mentioned earlier, our situation would qualify as both.
