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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

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  • $\begingroup$ What is your definition of curve? $\endgroup$ – Jack D'Aurizio Mar 29 at 13:55
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Whether a curve or surface can be parametrized would depend on how the curve or surface is given.

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 $f(x,y)=c$.

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One usually defines a curve/surface through a parametrization. So the existence is already build into the definition.

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