Smooth plane curves: definition

multivariable-calculus

My syllabus defines smooth plane curves as follows

A smooth curve in $\mathbb{R}^2$ is every subset $\Gamma$ of $\mathbb{R}^2$ that can be written as $\Gamma = \mathbf{r}[a,b]$, with $\mathbf{r}:[a,b] \to \mathbb{R}^2 \quad(a<b)$ a vector valued function with following characteristics:

  1. $\mathbf{r}$ is a bijection from $[a,b]$ to $\Gamma$
  2. $\mathbf{r} \in C^1[a,b]$
  3. $\mathbf{r}'(t) \ne \mathbf{0}$ for all $a \le t \le b$.

I have a few question regarding this definition (it's OK if you can only answer one of them, please do so down below):

  • $\mathbf{r} \in C^1[a,b]$ is a bijection $[a,b] \to \Gamma$. Does this imply that $\mathbf{r}^{-1} \in C^1(\Gamma)$ is a bijection that (locally) maps the curve to a straight line (in $[a,b] \subset \mathbb{R}$)?
  • Characteristic 3 in the definition means that the derivative of $\mathbf{r}$ with respect to $t$ cannot be the zero-$\underline{vector}$. Suppose $\mathbf{r}(t) = (x(t),y(t))$. Characteristic 3 actually states that $ \frac{dx}{dt}\mathbf{e}_x+\frac{dy}{dt}\mathbf{e}_y \ne \mathbf{0}$. Does this imply that one of the derivatives $x'(t), y'(t)$ can be zero, but not both of them?
  • I don't see why $\mathbf{r}'(a)$ and $\mathbf{r}'(b)$ cannot be equal to the zero-vector. Is it because $a$ and $b$ are boundary points? I believe we can take a restriction $\mathbf{r}_{\varepsilon}:[a+\varepsilon,b-\varepsilon] \to \mathbb{R}^2$, which (after $\varepsilon \to 0+$) will give us the same properties of the original curve (and $\mathbf{r}$). Isn't this the same as allowing $\mathbf{r}'(t)$ to be $\mathbf{0}$ in $a$ and $b$?

Thanks for answering and helping me out.

Best Answer

  • Yes, $\textbf{r}^{-1}$ maps the curve to the straight line (since the curve does not intersect itself and is regular)
  • Indeed, one component of the tangent vector: take, as an example, the curve $t\to (f(t,t))$, and look at its stationary points: there $\textbf{r}'(t_s)=(0,1)$.
  • Your extension does, indeed, preserve a certain amount of proprerties of the curve (a similar extension exists and is called, in fact, piecewise regular curve), but are usually a little bit more difficult to handle (consider, as an example, that if you join two curves non differentiable at their endpoints, the resultant curve will not be differentiable even in it's interior somewhere). Having a zero tangent vector means, naively, that there is a stop there (the velocity is zero): thus, after the stop you could go in every direction, not following any kind of regularity, and this creates some problems sometimes (see Frenet-Serrat's formula if you want to know more about this)
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