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My book states "... if $S$ is a level curve and $C$ is a curve in $S$ passing through a point $a$ ..."

What is a curve in $S$? Is it simply a subset of $S$?

From what I comprehend, a graph is usually a ordered set of the form $ X = \{(a,f(a))\ |\ a \in \mathrm{dom}(f)\}$

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up vote 3 down vote accepted

By "curve in $S$" your book means a restriction of $S$ which still represents a curve. For example, if we consider the curve described by $$ (x,y) = (\sin\theta, \cos\theta) $$ for $\theta \in [0, 2\pi]$, then an example of a curve "inside" this curve would be $$ (x,y) = (\sin\theta, \cos\theta) $$ for $\theta \in [0, \pi/2]$.

Probably the reason your book worded it this way is because it doesn't want to include every such restriction. For instance, restricting the domain to $[0,2\pi ] \cap \mathbb{Q}$ gives a subset of the curve, but not a "subcurve" (this terminology is not used as far as I know, but I didn't know what else to call it.)

Note that in introductory calculus courses often times there is no distinction made between a curve, the range of a curve, the graph of a curve, and the parametric representations of a curve. This is often times a source of confusion (as it may have contributed here), but the distinction should be properly handled in a more rigorous analysis setting.

You are correct that the graph of a curve is given by $\{(a,f(a)) | a \in \mathrm{dom}(f)\}$

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+1. Really nice answer. Thanks. Just 2 more questions: 1. Won't $[0,2\pi] \cap \mathbb{Q}$ be a sub-curve? Why not? 2. Can you point me towards a good online reference which summarizes the difference between "curve, range of curve, graph of curve, parametric representation of curve". –  Inquest Jul 7 '12 at 15:15
    
The restriction to $[0,2\pi]\cap \mathbb{Q}$ wouldn't be considered a curve by most people because curves should fit the intuitive notion of a curve. Namely we would want it to be continuous. –  nullUser Jul 7 '12 at 15:21
    
I'm still a bit confused. What is a curve? It's not a subset of a graph. Is it a continuous subset of a graph? –  Inquest Jul 7 '12 at 15:26
    
A parametric representation is just another word for a function which gives the connotation "we are about to be talking about curves". A curve is an equivalence class of parametric representations whereby two parametric representations are considered equivalent if there is a suitable (continuous, invertible, or more) change of variables from one to the other. The graph of a curve is given by $\{(a, f(a)): a \in \mathrm{dom}(f)\}$ where $f$ is a parametric representation of the curve. And the range of the curve is $\{f(a): a \in \mathrm{dom}(f)\}$. –  nullUser Jul 7 '12 at 15:27
    
Good enough for me as of now. –  Inquest Jul 7 '12 at 15:31
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