Is there a way in general to tell whether a given curve is parametrizable?
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Aren't all curves? – charlotte Jan 26 '15 at 05:06
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Many define curve as a class of parametrizations, sometimes even just one parametrization. This question might be a tautology. Maybe you are thinking in some implicit definition, as in the solutions to an equation? – Pp.. Jan 26 '15 at 05:06
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Sorry, I guess I don't really have a good handle on the subtleties of the different definitions of a curve yet. I think what I'm asking is, "Is there a way in general to tell when a given level curve, i.e. some general function in n variables, is parametrizable?" – Danny Jan 26 '15 at 05:10
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Yes, the idea is to use the implicit function theorem. Of course, we get the existence of local parametrizations this way. When the equations are polynomial, sometimes the implicit function theorem doesn't hold, but there is a very difficult to prove theorem that shows that you can parametrize. – Pp.. Jan 26 '15 at 05:12
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Thank you. Is there an online resource where I can read more about how this works? Right now I'm working out of Elementary Differential Geometry by Pressley but I haven't seen a characterization of parametrizable level curves yet. – Danny Jan 26 '15 at 05:14
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I tried googling for such a resource but I couldn't find one. I think the issue is that I don't know the name of the theorem that will give me whay I'm looking for. – Danny Jan 26 '15 at 05:17
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I am writing a more detailed answer. – Pp.. Jan 26 '15 at 05:19
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Suppose you are considering the level set $$f(x,y)=c$$ and we want to study whether we can parametrize this curve near the point $(x_0,y_0)$, which is a point in it, $f(x_0,y_0)=c$. If one of the partial derivatives is non-zero at that point, say $\frac{\partial f}{\partial y}(x_0,y_0)\neq0$ then we can use the Implicit function theorem to argue that there is such a local parametrization.
The implicit function theorem can be used for any number of variables.
Example: Suppose we study $$\sin(y)+y-xe^x=0$$ and we want to see if there is local parametrization at the origin $(0,0)$. We see that $$\frac{\partial}{\partial y}(\sin(y)+y-xe^x)|_{(0,0)}=(\cos(y)+1)|_{(0,0)}=2\neq0$$ Therefore, by the Implicit function theorem there is a function $g(x)$ such that $$\sin(g(x))+g(x)-xe^x=0$$ for all $x$ in a neighborhood of $x=0$.
When $f(x,y)$ is a polynomial sometimes the implicit function theorem is not applicable because the condition on the derivatives are not met.
Example: If we look at $x^2-y^3=0$ at the origin $(0,0)$ both partial derivatives are zero. Still, a parametrization exists $t\mapsto(t^3,t^2)$.
It is a very hard-to-prove theorem that in the case of polynomials nice parametrizations like this always exist.
