Smooth Functions

$\boldsymbol{r}(t)$ is smooth on an open interval $I$ iff:

• Differentiable
  $\boldsymbol{r}$ is differentiable on $I$, no corners, cusps, or breaks in the position function.
• Continuous derivative
  $\boldsymbol{r}'$ is continuous on $I$, the direction and speed of the curve can’t jump abruptly.
• Non-vanishing derivative
  $\boldsymbol{r}'(t) \ne \vec{0}$ for all $t \in I$. If the velocity vector hits zero, the curve can stop and reverse, or form a sharp point, even though $\boldsymbol{r}$ is still differentiable there.

A curve traced by $\boldsymbol{r}(t)$ may fail to be smooth where $\boldsymbol{r}'$ is not continuous or where $\boldsymbol{r}'(t) = \vec{0}$.

Examples:

• $\boldsymbol{r}(t) = \langle 6\cos t - \cos 6t, 6\sin t - \sin 6t \rangle$, $0 \le t \le 2\pi$, is not smooth at the points between its petals.