# Equation of a curve on a surface

1. Sep 20, 2013

### 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?

2. Sep 20, 2013

### Staff: Mentor

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. Sep 20, 2013

### LCKurtz

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. Sep 20, 2013

### Luna=Luna

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

Makes sense now.