# What are the conditions for a curve/surface to be parametrized?

I wanted to know what's on the title because I see a lot of people parametrizing curves/surfaces or saying that they can be parametrized somehow, but I never really saw a proof that it can really be done in the general case (from $$R^2$$ to $$R^n$$) (in particular cases, it's sufficient to see if the way the person parametrized is ok with the conditions of the curve/surface I think)

I'd really appreciate if someone could help me on this question

• What is your definition of curve? – Jack D'Aurizio Mar 29 at 13:55

The Implicit Function Theorem is relevant here; it gives sufficient conditions under which a condition $$f(x,y)=c$$ is (locally) equivalent to $$y=g(x)$$: that last is equivalent to a (local) parametrization $$x\mapsto (x, g(x))$$ of the level set $$f(x,y)=c$$.