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Description of surface, vector calculus

  1. Feb 14, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider the surface parameterized by (v cos(u), v sin(u), 45 v cos(u)), where u and v both vary from 0 to 2∏.


    2. Relevant equations
    (v cos(u), v sin(u), 45 v cos(u))
    I think this is supposed to be a vector function? As in r(u,v) = <v cos(u), v sin(u), 45 v cos(u)>.


    3. The attempt at a solution
    In the x-y plane, this is a circle. x = v cos(u) so z = 45x. This is a slanted plane? So I thought the surface would be an ellipse, since the coefficient of x is 45 and the circle would be very squashed. Does this seem to be part of a cylinder because the cross section of a cylinder, if the plane is slanted, would it be a portion of an elliptical paraboloid? Or could it be a portion of a cone or hyperboloid? One of the answer options is an ellipsoid, but I don't think that's right because when I graphed this on a computer, it showed a flat surface.

    There are so many possibilities!
     
    Last edited: Feb 14, 2013
  2. jcsd
  3. Feb 14, 2013 #2

    haruspex

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    Take v to be fixed, initially. Varying u should give you a curve, and yes, it's an ellipse. Describe the plane the ellipse lies in and where it is centred. Then imagine varying v over its given range.
     
  4. Feb 14, 2013 #3

    LCKurtz

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    Yes, that's exactly what it is and how you describe a surface.

    Yes, that is correct for fixed ##v##. Its equation is ##x^2+ y^2 = v^2##. But if you let ##v## vary from ##0## to ##2\pi## what do you get?

    Yes. And ##x## and ##z## must be on that plane, no matter what ##y## is.
    There aren't that many possibilities. You know all the points on the surface must be on the plane ##z=45x## and your computer shows the surface is flat. Don't those agree? If you project the ##xy## "shadow" in the ##xy## plane up onto the slanted plane, what do you get?
     
  5. Feb 17, 2013 #4
    So the description of the surface just an ellipse? If you project the circle onto the slanted plane, it should look like an ellipse.
     
  6. Feb 17, 2013 #5

    haruspex

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    I suppose you could describe it as an elliptical disc.
     
  7. Feb 18, 2013 #6
    Oh OK.
    Thank you for your help.
     
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