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Finding the unit normal of a vector field

  1. May 31, 2012 #1
    I'm on the last chapter of a 1200 page calc book, I'm really psyched.

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

    Screenshot2012-05-31at41711AM.png

    Screenshot2012-05-31at41418AM.png



    3. The attempt at a solution

    The method I learned for finding the unit normal of a vector field, n, is take the derivative of the equation and divide that by the magnitude of the derivative. This technique works for questions 3 and 4 above, but for 1,2 and 5, the solution manual just says n = k. I can't figure out why n = k. What are they talking about?
     
    Last edited: May 31, 2012
  2. jcsd
  3. May 31, 2012 #2
    I also need to know what d sigma = dxdy means.
     
  4. May 31, 2012 #3
    dσ=dx dy simply means that the differential of area is equal to the differential of x multiplied by the differential of y.
     
  5. May 31, 2012 #4
    Because the curve is parallel to the xy plane.
     
  6. May 31, 2012 #5

    sharks

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    [itex]\hat n = \vec k[/itex] means that the unit normal vector is parallel to the z-axis, in this case, it is in the vertically upward direction.
     
  7. May 31, 2012 #6
    what about d sigma?

    how do you find n if the curve is parallel to the xy plane, does it just equal k?
     
  8. May 31, 2012 #7
    Yes.
     
  9. May 31, 2012 #8
    d sigma is dx dy.
     
  10. May 31, 2012 #9
    Or in other words, the curve is parallel to the xy plane.
     
  11. May 31, 2012 #10

    sharks

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    n could be k or -k, depending on the orientation of the surface. n is the outward unit normal vector, and if the problem states "counterclockwise from above" then n is pointing upward, otherwise if it states ""counterclockwise from below"" then n = -k. If the surface lies in any plane (x=0, y=0 or z=0), then there are only 2 possible orientations for n, without any need for calculations, as you can deduce n directly.

    If the surface lies in:
    plane z=0, n is either k or -k
    plane y=0, n is either j or -j
    plane x=0, n is either i or -i

    In all 3 cases, the positive value denotes moving along the positive direction parallel the respective axis, and similarly the negative value denotes moving in the negative direction parallel to the axis.
     
    Last edited: May 31, 2012
  12. May 31, 2012 #11
    The OP said that the problem says n=k, so n is just k.
     
  13. May 31, 2012 #12
    but what does d sigma refer to? how do you calculate it? what does it mean?
     
  14. May 31, 2012 #13

    sharks

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    ##d\sigma## refers to the differential area, which when expressed in terms of Cartesian coordinates, converts to dxdy, as you are projecting the surface onto the x-y plane.
     
  15. May 31, 2012 #14
    As sharks has said, it is the differential of area.

    [tex]{\rm{d}}\sigma = \left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\|{\rm{d}}s{\rm{ d}}t[/tex]

    In your case, [itex]\left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\| = 1[/itex] and ##s=x,t=y##
     
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