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Math Help - some questions for the final

  1. #1
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    some questions for the final

    True or False. (why)

    1. There is a vector field F in R^3 such that curl(F)= y i + x j +z k. why?

    2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have \int\int_{\omega{1}}f dS =-\int\int_{\omega{2}}f dS   why?

    My thought: if f is scalar function, then this statement would be false?? Does this surface has to be a closed surface?

    3. For any smooth vector field F on a torus \omega \subset R^3 ,  \int\int_{\omega} curl(F)\cdot dS is zero
    My thought: use Stoke's theorem? omega has to be no boundary?

    4. Let F(x, y, z)=2x \imath - siny \jmath + z cosy \kappa and \sum_{R} be the sphere x^2 +y^2 + z^2 = R^2 oriented with outward-pointing normal. Then, the flux \int\int_{\sum_{R}}F \cdot dS increases with the radius R.
    Last edited by yzc717; May 16th 2009 at 08:56 PM.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by yzc717 View Post
    True or False. (why)

    1. There is a vector field F in R^3 such that curl(F)= y i + x j +z k. why?
    false. for a vector field F we must have \text{div}(\text{cur}(F))=0 but in here we have \text{div}(\text{cur}(F))=1.


    2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have \int\int_{\omega{1}}f dS =-\int\int_{\omega{2}}f dS why?
    true. properties of surface integral. see your lecture notes!


    3. For any smooth vector field F on a torus \omega \subset R^3 , \int\int_{\omega} curl(F)\cdot dS is zero
    true. the divergence theorem + see my answer to part 1 of your problem.


    4. Let F(x, y, z)=2x \imath - siny \jmath + z cosy \kappa and \sum_{R} be the sphere x^2 +y^2 + z^2 = R^2 oriented with outward-pointing normal. Then, the flux \int\int_{\sum_{R}}F \cdot dS increases with the radius R.
    true. by the divergence theorem the value of your surface integral is \frac{8}{3}\pi R^3.
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