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Finding total outward flux through a pyramid?

  1. Dec 13, 2009 #1
    The problem statement, all variables and given/known data
    Let F=xi-yj+zk and the pyramid be as shown below. Find [tex]\iint\limits \,[/tex][tex]F\bullet dS[/tex] as a surface integral over all five faces.

    The attempt at a solution
    I first solved it using the divergence theorem. The divergence was just 1, so the answer should just be the volume of the pyramid, which is 1/3, right?

    I think that the normal vectors of the two angled faces are [tex]i+k[/tex] and [tex]j+k[/tex], but I don't understand how to find the dS or the limits of integration since no real equation was given for the surface. Tried doing the surface integrals of the two angled faces assuming that the two angled faces were the intersection of the surfaces [tex]z=1-x[/tex] and [tex]z=1-y[/tex], but my answer is not coming out to 1/3. The flux through the non-angled faces (left, back, and bottom) should just be zero, right? I don't know whether I am just not seeing it or what, but can you guys help me out? Thanks in advance.
  2. jcsd
  3. Dec 13, 2009 #2


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    Science Advisor
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    Gold Member

    Yes, that looks correct.

    In the xy plane draw the diagonal from (0,0,0) to (1,1,0). That is the line y = x in the xy plane and the two triangles it forms lie directly under the two slanted faces. So if you express the integrals in terms of x and y, you can read the limits in the xy plane.

    As for calculating the vector dS, if you express the surface as a vector function R(x,y), using x and y as the independent variables you can use the formula

    [tex]\int\int_S \vec F \cdot d\vec S = \pm\int\int_R \vec F \cdot \vec R_x \times\vec R_y\,dxdy[/tex]

    where the sign is chosen so the direction of Rx X Ry agrees with the orientiation of the surface.

    For example, the front slanted plane x + z = 1 can be written:

    R(x,y) = < x, y, 1-x >

    Just use the formula above and express everything in terms of x. Read the limits from the xy plane. Similarly for the other slanted plane.

    That's right. I presume you know why, and if this is a homework problem, you should explain why.
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