Equation of a curve on a surface

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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?
 

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  • #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.
 
  • #3
LCKurtz
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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?
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.
 
  • #4
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Thanks for the responses!
I knew it had to be missing something basic!

Makes sense now.
 
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