Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)\}$

share|cite|improve this question
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 $\{(a,f(a)) | a \in \mathrm{dom}(f)\}$

share|cite|improve this answer
+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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.