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Math Help - Circulation on a velocity field

  1. #1
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    Circulation on a velocity field

    I have this exercise which asks me to calculate the circulation of a velocity field on V over the indicated path, this is it:

    \vec{V}=xy^2 \hat{i}+xe^{xy} \hat{j},y=x^2,x=0,y=1

    The thing is I don't know if I must use the path y=x^2 from (0,0) to (1,1), or if it reefers to three different paths with y=x^2,x=0,y=1 In that case I wouldn't know which extremes to use in the line integral. And the exercise that follows its similar, it gives a path, but it don't tells from where to where, but I think that in that case could be because the path is closed.

    Bye and thanks!
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  2. #2
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    Yes, "circulation" must occur on a closed path. This is actually a single path with three different parts. Draw the three graphs given and you should see which closed path this is referring to: the flow along y= x^2 from (0, 0) to (1, 1), along y= 1 from (1, 1) to (0, 1) and along x= 0 from (0, 1) to (0, 0) again.

    You could also use Green's theorem to do this. In fact, it would be good idea to do it both ways as a check!
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Ulysses View Post
    y=x^2,x=0,y=1

    The above expression is ill defined. Perhaps someone meant:

    "The piece \gamma of the parabola y=x^2 from x=0 to x=1".

    Or equivalently:

    \gamma:\; \begin{Bmatrix} x=t\\y=t^2\end{matrix}\quad t\in{[0,1]}

    Edited: Sorry: I didin't read "circulation". My observation was only to the way of expressing the curve.

    Regards.
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