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Homework Help: Integrals with curl dot products

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

    1. Evaluate [tex]\int_{S}\int curl F \cdot N dS[/tex] where S is the closed surface of the solid bounded by the graphs of x = 4, z = 9 - y^2, and the coordinate planes.

    F(x,y,z) = (4xy + z^2)i + (2x^2 + 6y)j + 2xzk

    2. Use Stokes's Theorem to evaluate [tex]\int_{C}F\cdot T dS[/tex]

    F(x,y,z) = xyzi + yj +zk
    S: 3x+4y+2z=12, first octant

    2. Relevant equations

    3. The attempt at a solution

    1. For this one, I found the curl to be -6yi. However, I am at a loss as to how to get the N dS part without some sort of given equation for S? The book answer is 0.

    First I found the curl to be:
    [tex]xyj - xzk[/tex]

    I then used a theorem in my book to find N ds:
    3/2i + 2j + k

    Then I took the dot product:
    [tex]<0, xy, -xz> \cdot <\frac{3}{2}, 2, 1> = 2xy - xz[/tex]

    [tex]\int^{4}_{0}\int^{4-\frac{4y}{3}}_{0}(2xy-x(-\frac{3x}{2} - 2y + 6)*dx*dy [/tex]
    which comes out to 64/27.

    The book answer is 0.

    Any pointers as to what I'm doing wrong would be appreciated.
  2. jcsd
  3. May 1, 2010 #2


    User Avatar
    Homework Helper
    Gold Member

    You can either use Stokes' theorem for this, and the answer will be immediate; or you can find the equation of the surface that bounds the volume described in the question and integrate directly. (If you are going to use the second method, I recommend you break the surface into 4 separate surfaces to make it easier, and start by sketching the volume so you can see what I mean)

    Shouldn't [itex]\textbf{n}dS[/itex] be a differential vector?:wink:
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