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Parametrized Surface Problem

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data
    If a regular surface can be parametrized in the form

    [tex]s(u, v) = a(u) + b(v)[/tex]

    where a and b are regular parametrized curves with [tex](u,v)[/tex] in some domain [tex]D\subseteq\mathbb{R}^2[/tex], show
    that the tangent planes along a fixed coordinate curve of [tex]D[/tex] are all parallel to a line.


    2. Relevant equations
    The normal to the tangent plane at [tex](u_0,v_0)[/tex] is given by

    [tex]N(u_0,v_0)=\partial_1s(u_0,v_0)\times\partial_2s(u_0,v_0)[/tex]

    where

    [tex]\partial_is=\frac{\partial s}{\partial x_i}[/tex]


    3. The attempt at a solution
    Taking my fixed coordinate curve in [tex]D[/tex] to be

    [tex]L:=\{(u_0,t) | t\in\mathbb{R}\}[/tex]

    I found the normal using the equation above to be

    [tex]N(t)=a'(u_0)\times b'(t)[/tex]

    But a basis for the [tex]b'(t)[/tex]'s is at most 3 dimensional, for [tex]N(t)[/tex] to be orthogonal to a line (and thus the tangent plane to be parallel to a line) a basis for [tex]N(t)[/tex] would have to be 2 dimensional. So I cannot show the statement of the problem. Am I doing something wrong, is there a better way to approach the problem?
     
  2. jcsd
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