# Equation of a curve on a surface

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
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.

LCKurtz
Homework Helper
Gold Member
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.

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

Makes sense now.