Results 1 to 2 of 2

Thread: some questions for the final

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    44

    some questions for the final

    True or False. (why)

    1. There is a vector field F in $\displaystyle R^3$ 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 $\displaystyle \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 $\displaystyle \omega \subset R^3 , \int\int_{\omega} curl(F)\cdot dS$ is zero
    My thought: use Stoke's theorem? omega has to be no boundary?

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

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by yzc717 View Post
    True or False. (why)

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


    2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have $\displaystyle \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 $\displaystyle \omega \subset R^3 , \int\int_{\omega} curl(F)\cdot dS$ is zero
    true. the divergence theorem + see my answer to part 1 of your problem.


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

Similar Math Help Forum Discussions

  1. Final Questions for Practice Test
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Nov 3rd 2009, 02:26 PM
  2. Inductive Questions, final algebra part
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Oct 15th 2009, 07:02 PM
  3. Sample Questions for final
    Posted in the Statistics Forum
    Replies: 13
    Last Post: Sep 2nd 2008, 08:05 PM
  4. 2 final trig questions LOL ( then i'm done )
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 26th 2007, 12:35 AM
  5. Important Questions (Preparing For Final) Need Help
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: May 14th 2006, 05:29 AM

Search Tags


/mathhelpforum @mathhelpforum