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Math Help - Flux integrals - verifying Stokes Theorem

  1. #1
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    Flux integrals - verifying Stokes Theorem

    Ok, I've been stuck on this problem for hours now and its really irritating, so I need help!

    Here is the question:

    -------------------------------------------------------------

    Verify Stokes Theorem for the vector field

    F = yi + 2zj + xzk

    and the surface S defined by

    x^2 + y^2 + z^2 = 25 & Z>=4

    --------------------------------------------------------------

    Right, I'm quite new to these so please explain clearly if possible, cheers!

    So far I've computed the line integral and got a result of -Pi

    Now I'm stuck on the flux integral. I've worked out that Curl F = -2i -zj -k but I'm not sure what the normal vector is or the limits.

    Any help much appreciated!
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  2. #2
    MHF Contributor Calculus26's Avatar
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    The surface is the portion of the hemi -sphere x^2 +y^2 +z^2 =25

    above z= 4

    z= sqrt(25-x^2 -y^2)

    N = -dz/dx i -dz/dy j + K

    N = x/sqrt(25-x^2) i + y /sqrt(25-y^2) j + k

    The region of integration is the circle x^2 + y^2 = 9

    (using z=4)

    See the attachment for the calculation of the line integral using both Stokes Theorem and the definition of line integral
    Attached Files Attached Files
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  3. #3
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    Gah so my line integral result is incorrect?
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  4. #4
    MHF Contributor Calculus26's Avatar
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    you're computing the line integral around the circle of radius 3 , 4units

    above the x-y plane.

    How'd you compute the line integral ?

    What was your parameterization and F*dr/dt ?
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  5. #5
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    I had circle radius 1, 4 units

    My parameterization was r(t) = cos(t)i +sin(t)j +4k for 0<t<2Pi

    F.dr = -sin^2(t) + 8cos(t)
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  6. #6
    MHF Contributor Calculus26's Avatar
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    x^2 + y^2 +z^2 = 25

    if z = 4 x^2 + y^2 = 9

    Which gives x=3cos(t) y = 3sin(t) z = 4

    r ' = -3sin(t) i +3cos(t) j

    F = 3sin(t) i - 4 j + 12cos(t) k

    As in the attachment. you then end up with -9pi which is the - 28.274 in the attachment
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  7. #7
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    How did you go about doing the first integral, ie "we obtain using the parameterization". In the word document i meant
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  8. #8
    MHF Contributor Calculus26's Avatar
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    The first integral is the integral curl F*N

    Then using the pararmaterization for the bounding curve we obtain the second integral of F*dr/dt
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