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Math Help - Vector Calculus: Divergence theorem and Stokes' theorem

  1. #1
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    Vector Calculus: Divergence theorem and Stokes' theorem

    Hi =)
    I've been working on some exercises, but unfortunately we are not given the answers, so have no idea if I am going wrong or not. I attemted part a) which reads:
    Consider the vector field
    V= 3x^2y i - 2xy^2 j - xyz k

    Determine the divergence of this vector field and evaluate the integral of this quantity over the interior of a cube of side a in z> or equal to 0, whose base has the vertices (0,0,0), (a,0,0), (0,a,0), (a,a,0).
    Evaluate the flux of V over the surface fo the cube and thereby verify the divergence theorem.


    So for this I got both sides to be 1/4 a^5

    Its the next part that has got me completely confused! (its long, here goes..! if anyone can teach me how to use the integral symbols and stuff, i will edit this post to make more sense!)

    [B]b) Given V = (3x-2y) i + x^2z j + (1-2z) k
    Find del.V and del x V and evaluate:[B]
    I found them to be (3x-2y) i + x^2y j + (1-2z)k
    and -x^2 i + (2xz-2) k respectively.

    [B]i) double integral of V.dS
    over the circular region in the xy-plane bounded by x^2 + y^2 = a^2. Regard the area element as positive in the positive z direction.[B]

    What is dS supposed to be?!

    ii) Evaluate
    the double integral of (del x V) . dS
    over the same region.

    iii) Evaluate
    the closed loop integral of V.dr
    clockwise looking in the positive z direction around the circle x^2 + y^2 = a^2 in the xy-plane.

    iv) Evaluate
    the triple integral of (del x V)dV
    over the volume of the hemisphere bounded by the spherical surface x^2 + y^2 + z^2= a^2 for z>0 and the xy-plane.

    Which two answers provide a verification of Stokes' Theorem?


    Thank you SO much
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  2. #2
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    Can no one help me?
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  3. #3
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    first you can quote other's post, so that you can see the source of the LaTex content. It's only a simple tag "math" warrping the LaTex source code.

    V= 3x^2y i - 2xy^2 j - xyz k

    and dS is the volume element vector.
    \int V.dS is the surface integral with V.n as the integrand, where n is the outer pointing unit normal vector of the surface.

    And I think if your results (surface integral and volume integral) agree, they are supposed to be both correct.
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