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Application of Stokes' theorem

  1. Jul 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following integrals

    [tex]I_1 = \oint \vec{r} (\vec{a} \cdot \vec{n}) dS[/tex]
    [tex]I_2 = \oint (\vec{a} \cdot \vec{r})\vec{n} dS[/tex]

    where [tex]\vec{a}[/tex] is a constant vector, and [tex]\vec{n}[/tex] is an unit vector normal to the closed surface [tex]S[/tex].


    2. Relevant equations

    Stokes' theorem, or in this case, some sort of Gauss' divergence theorem.

    3. The attempt at a solution

    I'm not sure whether the Stokes's theorem can be used in this case. For instance, if we write the integrals in the component notation

    [tex]I_1 = \oint x_i (a_j dS_j)[/tex]
    [tex]I_2 = \oint (a_j x_j)dS_i[/tex]

    then the Stokes' theorem would suggest the supstitution [tex]dS_i \to dV \partial_i[/tex] where [tex]S=\partial(V)[/tex]. This results in

    [tex]I_1 = \int (a_j \partial_j) x_i dV=\vec{a} V[/tex]
    [tex]I_2 = \int \partial_i (a_j x_j)dV=\vec{a} V[/tex]

    Is this correct?
     
  2. jcsd
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