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Zero curl but nonzero circulation

  1. Dec 31, 2012 #1
    The vector field [itex]\vec{F} = <\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0>[/itex] has a zero curl, which means its circulation is zero. However

    [itex]\int \vec{F}.d\vec{s}[/itex] around a unit circle on the xy plane is equal to [itex](+/-)2\pi[/itex] and not zero

    Is it because F is undefined at (0,0)? No, because Stoke's theorem allows me to choose an arbitrary surface not including the origin(0, 0, 0)?
    Last edited: Dec 31, 2012
  2. jcsd
  3. Dec 31, 2012 #2


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    Right - more general, it is undefined at (0,0,z). As result, the domain of F is not simply connected - and the surface you want to consider does not exist.
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