Equation of a curve on a surface

[Luna=Luna]

Trying to understand a concept on vector calculus, the book states:
If S is a surface represented by

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

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?

 

[mfb]

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.

 

[LCKurtz]

Trying to understand a concept on vector calculus, the book states:
If S is a surface represented by

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

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.

 

[Luna=Luna]

Thanks for the responses!
I knew it had to be missing something basic!

Makes sense now.