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Math Help - calculating flux

  1. #1
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    calculating flux

    A vector field  \vec{G} in 3-space is defined outside the cylinder  x^2 + y^2 = 4


     \vec{G} = \frac{6y\vec{i}-6x\vec{j}}{x^2+y^2}

    Find  \int\limits_C \vec{G} \cdot   d\vec{r} where C is the circle x^2 + y^2 = 16 in the xy-plane and oriented counterclockwise when viewed from above.

    I am planning to use green's theorem to solve this problem and have found the curl to be:

    \frac{12x^2}{x^2+y^2) is this true?? Then I need to parametrize the circle.. and integrate it..
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  2. #2
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    Quote Originally Posted by TheRekz View Post
    A vector field  \vec{G} in 3-space is defined outside the cylinder  x^2 + y^2 = 4


     \vec{G} = \frac{6y\vec{i}-6x\vec{j}}{x^2+y^2}

    Find  \int\limits_C \vec{G} \cdot  d\vec{r} where C is the circle x^2 + y^2 = 16 in the xy-plane and oriented counterclockwise when viewed from above.

    I am planning to use green's theorem to solve this problem and have found the curl to be:

    \frac{12x^2}{x^2+y^2) is this true?? Then I need to parametrize the circle.. and integrate it..

    the parameterization of the circle is \vec r(t) =4\cos(t) \vec i +4\sin(t) \vec j

    The direct computation is quick

    \vec r'(t) =-4\sin(t) \vec i +4\cos(t) \vec j

    \int\limits_C \vec{G} \cdot  d\vec{r} =\int_{0}^{2\pi}\frac{6(4\sin(t)(-4\sin(t))-6(4\cos(t)(4\cos(t)))}{16}dt

    -6\int_{0}^{2\pi} \sin^2(t)+\cos^2(t)dt=-6\int_{0}^{2\pi}dt=-12\pi
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