Maximum Number of Closed Curves with zero Line Integral

Hi All,

I have been battling with this question for a while. Given a conservative vector field, we know
that there are infinitely many closed paths where the line integral evaluated is zero. In fact this is the requirement for a conservative vector field: Every line integral of any closed path is zero. Now let's take any non-conservative vector field. Could we say something about how many closed curves have a zero line integral? Of course the number should be less than infinity(otherwise it would be conservative!). But I was wondering if there is more to say(like an upper bound) or even say there is no such closed curve with line integral zero(although I highly doubt that). If you are confused on the above details
think the question below

Take a vector field F. Say I find, by some means, the line integral around any 1000 closed random paths is zero. Will this say anything whether the field is conservative or not?

Thanks a lot
Abiy
p.s books or papers along this topic will be appreciated

Office_Shredder
Staff Emeritus