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Given the parameterization of a curve in $\mathbb{R}^n$,

$$\boldsymbol \gamma (t) = (x_1(t), x_2(t), \ldots, x_n(t))$$

I can not find a univocal definition of smoothness.

This answer requires existing and non-zero first derivatives of the components of $\boldsymbol \gamma(t)$.

This post does not provide a definitive choice.

In this ProofWiki page (linked from the smooth curve page) instead both versions are provided:

Version 1) a function is smooth if it belongs to class $C^{\infty}$;

Version 2) a function is smooth if it has continuous first derivatives everywhere in its domain.

Therefore:

  1. Is there an unambiguous way to define smoothness, or is it an arbitrary concept, which may depend on the textbook, the author and/or his/her needs?

  2. Assuming that all the components $x_i(t)$ (with $i = 1, 2, \ldots, n$) of $\boldsymbol \gamma (t)$ must belong to class $C^{\infty}$, does this imply that their $k$-th derivative must also be non-zero? In other words: consider the simple example parameterization of a straight line in $\mathbb{R}^1$ $\boldsymbol \gamma (t) = t$. It has existing but zero derivatives starting from the order $k = 2$. Can this curve be considered smooth?

BowPark
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1 Answers1

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  1. Yes, it is an ambiguous terminology going back to the 18th century, which is why (modern) advanced math books tend to avoid it and instead talk about diffeomorphisms, local diffeomorphisms, covering spaces, etc. Yes, the usage of the terminology "parameterization" (for curves) varies from source to source. However, the terminology is not arbitrary, all (known to me) definitions of a parameterization have the following commonalities:

(a) A parameterization $\gamma$ (defined on a finite or infinite interval $I$ which also can be closed or open or half-open) is onto the target curve $\Gamma$.

(b) The map $\gamma$ is at least continuous. (Most sources will assume some degree of differentiability.)

(c) The map $\gamma$ is at least locally injective, i.e. every point $t_0\in I$ has a neighborhood $U$ in $I$ such that the restriction of $\gamma$ to $U$ is 1-1. (You will likely learn (or already learned) the definition of a neighborhood in a real analysis or a topology class.) For instance, "space-filling curves" will not be parameterizations. Local injectivity is implied by the regularity of a curve, which means that $\gamma$ is $C^1$ and has nonvanishing 1st derivative.

  1. As you correctly noticed, nonvanishing of derivatives of orders $\ge 2$ is not assumed and will fail even in simplest examples.
Moishe Kohan
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