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Equation of a curve on a surface

  1. Sep 20, 2013 #1
    Trying to understand a concept on vector calculus, the book states:
    If S is a surface represented by

    [tex]\textbf{r}(u,v) = u\textbf{i} + v\textbf{j} + f(u,v)\textbf{k}[/tex]

    Any curve r(λ), where λ is a parameter, on the surface S can be represented by a pair of equations relating the parameters u and v, for example u = f(λ) and v = g(λ).

    What exactly is the justification or proof for this statement?
  2. jcsd
  3. Sep 20, 2013 #2


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    The surface has exactly one point for each pair of (u,v) (in the range where the surface is defined), as f(u,v) has exactly one value for all (u,v).
    To define a curve on the surface, it is sufficient to specify a set of (u,v)-pairs.
  4. Sep 20, 2013 #3


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    Not a good idea to use ##f## for two different things. If ##u=h(\lambda),~v=g(\lambda)## then $$\vec r(u,v)=\langle h(\lambda),g(\lambda),f(h(\lambda),g(\lambda))\rangle$$which is obviously on the surface and is a parametric function of a single variable, so is a curve.
  5. Sep 20, 2013 #4
    Thanks for the responses!
    I knew it had to be missing something basic!

    Makes sense now.
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