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Parameterization, Folium of Descartes, etc.

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the surface z = f[itex]\left(x y\right)[/itex] = xy. Given a point P = [itex]\left[a, b, ab\right][/itex] on this surface, show that the lines with direction vectors u = [itex]\left[1, 0, b\right][/itex] and v = [itex]\left[0, 1, a\right][/itex] through P are entirely contained in the surface.

    2. The attempt at a solution

    To be honest, I'm not really sure what I'm supposed to do here or how to go about doing it, if someone could explain it and/or give me a starting point, that'd be very helpful.

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

    Consider the curve given by x3 + y3 = 3axy.

    What happens as t[itex]\rightarrow[/itex]-1? Consider the limit as t[itex]\rightarrow[/itex]-1 of x(t) + y(t). What do you conclude?

    The curve has obvious symmetry. Verify this using your parametrization.

    2. Relevant equations

    The parametrized equations are:

    x = 3at / 1 + t3

    y = 3at2 / 1 + t3

    2. The attempt at a solution

    For the limit, after adding and simplifying the equations, it equals -a. I believe the equation is asymptotic at t = -1, but I guess I'm not quite sure what I'm supposed to conclude or what -a means.

    For the symmetry part, it is symmetric about y = x, so that for every point (x, y) there is a point (y, x). How do I show this with the parametric equations?
     
  2. jcsd
  3. Sep 12, 2012 #2
    Sorry for the poor formatting. I don't know how to format column vectors or fractions.
     
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