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Math Help - Stokes Theorem

  1. #1
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    Stokes Theorem

    F = 3yi - 2yj + {z}^{2 }k S the hemisphere {x}^{2 } + {y}^{2 } + {z}^{2 }=1  z\geqslant 0  So i ahve done \nabla x F = 0i + 0j -3k Now my integral is \iiint -3{r}^{ 2} sin\phi dr d\phi d\theta where r is between 1 and 0, d\phi is between \frac{\pi }{ 2} and 0 and d\theta is between 2\pi and 0

    But this isnt gettign me the right answer what have i done wrong
    Last edited by adam_leeds; May 9th 2011 at 07:07 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Are you sure that your question is related to the Stokes Theorem?
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    Are you sure that your question is related to the Stokes Theorem?
    Verify Stokes theorem in the following cases
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  4. #4
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    The answer is -3\pi by the way
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  5. #5
    Behold, the power of SARDINES!
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    Quote Originally Posted by adam_leeds View Post
    F = 3yi - 2yj + {z}^{2 }k S the hemisphere {x}^{2 } + {y}^{2 } + {z}^{2 }=1  z\geqslant 0  So i ahve done \nabla x F = 0i + 0j -3k Now my integral is \iiint -3{r}^{ 2} sin\phi dr d\phi d\theta where r is between 1 and 0, d\phi is between \frac{\pi }{ 2} and 0 and d\theta is between 2\pi and 0

    But this isnt gettign me the right answer what have i done wrong
    You seem to be mixing Stokes Theorem with the divergence theorem.

    Given a Vector field and a surface we know that

    \iint_{S}\nabla \times \mathbf{F} \cdot d\mathbf{S}=\oint_{\partial S}\mathbf{F}\cdot d\mathbf{s}

    The boundary of this curve is the unit circle in the xy plane.

    My question is what is the integral you started with and what are you trying to compute? Are you trying to compute the flux of

    \mathbf{F}

    though the hemisphere?

    Edit: Just typing too slow
    Last edited by TheEmptySet; May 9th 2011 at 07:17 AM. Reason: too slow
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    Verify Stokes theorem in the following cases

    Then, what has to do the triple integral?. You have to prove


    \displaystyle\int_C \vec{F}\cdot d\vec{r}=\displaystyle\iint_{S}(\textrm{rot} \vec{F})\cdot \vec{n}\;dS


    Edited: Sorry, I didn't see TheEmptySet's post.
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  7. #7
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    Aaaahhh right, so diveregnce you can use a triple, stokes you cant?
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  8. #8
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    Quote Originally Posted by adam_leeds View Post
    Aaaahhh right, so diveregnce you can use a triple, stokes you cant?
    Still a bit stuck

    \iint -3{z}^{ 2} z = r so \iint -3{r}^{ 2}  r dr d\theta

    Is that right?
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  9. #9
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adam_leeds View Post
    Aaaahhh right, so diveregnce you can use a triple, stokes you cant?

    Yes, the Divergence Theorem states

    \displaystyle\iint_{S} \vec{F}\cdot \vec{n}\;dS=\displaystyle\iiint_{V}\textrm{div}( \vec{F})\;dxdydz
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  10. #10
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    Quote Originally Posted by adam_leeds View Post
    Still a bit stuck

    \iint -3{z}^{ 2} z = r so \iint -3{r}^{ 2}  r dr d\theta

    Is that right?
    These aren't spherical co-ordinates, how do i use spherical co-ordinates?
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  11. #11
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    Use the divergence theorem as hinted by others. The volume you're dealing with is a hemisphere which is easy to deal with in spherical coordinates.

    Spherical coordinate system - Wikipedia, the free encyclopedia



    The limits for a hemisphere with radius=1 is:
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  12. #12
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    Quote Originally Posted by Mondreus View Post
    Use the divergence theorem as hinted by others. The volume you're dealing with is a hemisphere which is easy to deal with in spherical coordinates.

    Spherical coordinate system - Wikipedia, the free encyclopedia



    The limits for a hemisphere with radius=1 is:
    Is \nabla \cdot F =  -2 + 2z
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