Maximum Number of Closed Curves with zero Line Integral

  • Thread starter abiyo
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  • #1
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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
 

Answers and Replies

  • #2
Office_Shredder
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There can be infinitely many closed curves for which the integral is zero for a non-conservative vector field. As long as there is any region in which the vector field is conservative, every closed loop in that region has an integral of zero over it
 
  • #3
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There can be infinitely many closed curves for which the integral is zero for a non-conservative vector field. As long as there is any region in which the vector field is conservative, every closed loop in that region has an integral of zero over it

Hi Office_Shredder,

Could you point me to a theory or proof for what you mentioned above?

Thanks
 

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