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Homework Help: Unit Normal Vector to a Cylinder

  1. Dec 15, 2009 #1
    1. The problem statement, all variables and given/known data
    I need to find the unit normal vector to x2+z2=16. I can find the unit normal vector to every shape but a cylinder, and I can't seem to see how to do it.

    2. Relevant equations

    3. The attempt at a solution
    I know that I need to find the gradient of the surface equation. I've considered that this gradient may be i-2x-2z. Or am I completely wrong?

    If you need more information please say what. Thank you!
  2. jcsd
  3. Dec 15, 2009 #2


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    You aren't really using your own "Relevant Equation". The level surface equation for your cylinder is F(x,y,z)=x^2+z^2-16=0. Now what is grad(F) again?
  4. Dec 15, 2009 #3
    So if the equation of the cylinder is x^2+z^2=16, i would f(x,y,z)=x^2+z^2, and find the gradient from there.

    Then you also have the constraint that it has to be on the cylinder x^2+z^2=16. So I would parametrize that surface, and then plug that parametrization into the gradient to get the equation for the vectors. Then you need to divide by the norm in order to get unit vectors, (which i think might end up very nicely!).

    Alternatively, the way i would do it, is just parametrize the cylinder off the bat, then take the derivative, then normalize it (but one doesn't use gradients directly).
  5. Dec 15, 2009 #4
    Wow, that was a terrible oversight on my part. I believe I took that from the other question I was looking at at the time. This gradient would be 2xi+2zk, no?
  6. Dec 15, 2009 #5
    yeep! looks like it!
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