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Flux of vector fields

  1. Dec 2, 2017 #1

    yecko

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

    Example 2:

    image.jpg image.jpg
    2. Relevant equations
    Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y)

    3. The attempt at a solution
    Why do my attempt is wrong? The example is using the foundational formula while I use the stock formula from the book, why is there a negative sign difference between the answers? Or is that my formula used inappropiately?

    Thanks!

    image.jpg
     
  2. jcsd
  3. Dec 3, 2017 #2

    LCKurtz

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    One reason PF discourages the use of images is that they are difficult to edit. On your first line you have the equation ##g = y = x^2##, whatever that means. You are likely using the formula for a surface of the form ##z = g(x,y)##. The surface ##y = x^2## is not that kind of surface because ##y## and ##x## are not independent. The easiest way to represent the surface is ##y = g(x,z)##. In any case, however you did it, your normal vector is in the wrong direction. The ##y## component of your normal vector must be negative.
     
    Last edited: Dec 4, 2017
  4. Dec 4, 2017 #3

    yecko

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    Thanks for pointing out the problem...
    but how can we see the direction of normal vector in this formula?
    and how to correct it? (simply by adding a negative sign???)
     
  5. Dec 4, 2017 #4

    LCKurtz

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    I can't tell how you got your normal or what formula you used because you didn't show your work. What I would do is parameterize the surface like this$$
    \vec R(x,z) = \langle x, x^2, z\rangle$$and get a normal by ##\vec R_x\times \vec R_z## and take it or its opposite, whichever has a negative ##y## component.
     
  6. Dec 4, 2017 #5

    yecko

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    Alright! I believe I've got it! thanks!
     

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