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## Main Question or Discussion Point

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

What exactly is the justification or proof for this statement?

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?