Curve
Definition 1. We call the vector function $\textbf{r}: (a, b) \rightarrow \mathbb{R}^3$ a curve. We can parametrize curves by $\textbf{r}(t) = (f(t), g(t), h(t))$ where $t \in (a, b)$.
Velocity, Speed, Unit Tangent Vector
Definition 2. Let $\textbf{r}$ be a curve. Then
(1) $\textbf{v}(t) = \frac{d \textbf{r}}{dt}$ is the velocity vector of $\textbf{r}$,
(2) $|\textbf{v}|$ is the speed of $\textbf{r}$,
(3) $\textbf{T} = \frac{\textbf{v}}{|\textbf{v}|}$ is the unit tangent vector.
Limit and Continuity of Curves
Definition 3. Let $\textbf{r}(t) = (f(t), g(t), h(t))$ be a curve with domain $D$, and let $\mathbf{L} \in \mathbb{R}^3$. We say that $\mathbf{r}$ has limit $\mathbf{L}$ as $t$ approaches $t_0$ and write $$\lim_{t \to t_0} \mathbf{r}(t) = \mathbf{L}$$ if, for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $t \in D$ $$|\mathbf{r}(t) - \mathbf{L}| < \epsilon \text{ whenever } 0 < |t-t_0| < \delta.$$
Definition 4. A curve $\textbf{r}(t)$ is continuous at a point $t = t_0$ in its domain if $$\lim_{t \to t_0} \textbf{r}(t) = \textbf{r}(t_0).$$ The curve is continuous if it is continuous at every point in its domain.
Differentiability of Curves
Definition 5. The curve $\text{r}(t) = (f(t), g(t), h(t))$ has a derivative (is differentiable) at $t$ if $f, g$ and $h$ have derivatives at $t$. The derivative is the curve $$\textbf{r}'(t) = \frac{d \textbf{r}}{dt} = \lim_{\Delta t \to 0} \frac{\textbf{r}(t + \Delta t) - \textbf{r}(t)}{\Delta t} = ( f'(t), g'(t), h'(t)).$$ The curve is differentiable if it is differentiable at every point of its domain.
다시 말해 곡선의 극한, 연속, 미분가능성은 component-wise하게 정의할 수 있다. 때문에 각 component function인 scalar function들이 극한을 가지는지, 연속인지, 미분가능한지 체크해서 취합해주면 된다.
Smooth
Definition 6. The curve $\textbf{r}$ is smooth if the velocity vector, i.e., $\frac{d \textbf{r}}{dt}$ is continuous and never $\textbf{0}$.
속도가 영벡터가 된다면 곡선을 따라 움직이는 입자가 멈췄다가 다시 움직인다고 물리적으로 해석할 수 있다. 그러면 말그대로 부드럽지 않고 방향을 전환하여 곡선이 꺾이는 등의 불상사가 발생할 수 있다. 때문에 이러한 상황이 발생하지 않는 곡선을 smooth하다고 말한다.
Length
Definition 7. The length of a smooth curve $\textbf{r}(t)$, $a \leq t \leq b$, that is traced exactly once as $t$ increases from $t=a$ to $t=b$, is $$L = \int_a^b |\textbf{v}| dt.$$
곡선의 '길이'로 생각할 수도 있고, 물리적으로는 곡선을 따라 움직이는 입자의 '이동거리'라고 생각할 수 있다.