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Homework Help: Surface Integrals: Flux of F across S

  1. Nov 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.

    F(x, y, z) = xy i + yz j + zx k

    S is the part of the paraboloid z = 3 - x[tex]^{2}[/tex] - y[tex]^{2}[/tex] that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation.

    2. Relevant equations
    [tex]\int \int F\cdot dS = \int \int \left( -P \frac{\partial g}{\partial x} -Q \frac{\partial g}{\partial y} +R \right) dA[/tex]

    3. The attempt at a solution
    I've gone over the examples in my calculus book two or three times now and I get confused about a couple things:

    dA is r*dr*d[tex]\theta[/tex], and I replace the x's and y's with rcos[tex]\theta[/tex] and rsin[tex]\theta[/tex], respectively, and substituting z with the given equation, but I still get a wrong answer. Here's my attempt:

    [tex]\int \int ( -y(-2x) -x(-2y) +3-x^{2}-y^{2})dA[/tex]

    ...and after substituting x and y and z for their polar coordinates and then simplifying, I get:

    [tex]\int ^{\pi / 2}_{0} \int ^{1}_{0} (3+4r^{2}cos(\theta)sin(\theta)-r^{2})rdrd\theta[/tex]

    I was kind of hoping this would get me the right answer, but it's not (I end up with [tex] \frac{5 \pi}{8} + \frac{1}{2}[/tex]), and I think it has to do with the r domain that I used, or perhaps my entire equation. In any case I'm not sure how to implement the square of length 1 in xy plane domain into my equation. Where have I gone wrong?

    Thanks in advance
  2. jcsd
  3. Nov 30, 2009 #2


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    This is your P(x,y)i + Q(x,y)j + R(x,y)k

    This is your z = g(x,y)

    Are your P(x,y), Q(x,y) and R(x,y) terms here correct?

    Why in the world would you try polar coordinates at this point? The integrand is simple polynomial terms in x and y and you are integrating over a square.​
    Last edited: Nov 30, 2009
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