### Differential geometry notes 2

Definition 9.1: Let $\mathbf{x}(t)$ $(a \leq t \leq b)$ be an allowable parametric representation of an arc $C$ with initial point $A$ and terminal point $B$. Denote by $l(Z)$ the length of a broken line of chords $Z$ whose end points lie on $C$ and correspond to the values $a = t_0 < t_1 < \ldots < t_n = b$. Let $n \rightarrow \infty$ and $\sigma(Z)= \underset{1 \leq v \leq n}{\max}(t_v - t_{v-1}) \rightarrow 0$. If $l(Z)\rightarrow s$, then $C$ is called rectifiable and $s$ is called the length of $C$.

Theorem 9.1: Let $\mathbf{x}(t)$ $(a \leq t \leq b)$ be an allowable parametric representation of an arc $C$ of a curve of class $r$ ($r$ times differentiable). Then $C$ has length:

$s = \int_{a}^{b} \sqrt{(\frac{dx_1}{dt})^2 + (\frac{dx_2}{dt})^2 + (\frac{dx_3}{dt})^2} dt = \int_{a}^{b} \sqrt{\mathbf{x^{\prime}} \cdot \mathbf{x^{\prime}} } dt$

and $s$ is independent of the choice of the allowable parametric representation.

Arc length: The function $s(t) = \int_{t_0}^{t} \sqrt{\mathbf{x^{\prime}} \cdot \mathbf{x^{\prime}} } dt$ is called the arc length of $C$.

The arc length may be used as the parameter in a parametric representation of a curve: $\mathbf{x}(s)$ is an allowable parametric representation. $s$ is called the natural parameter. The choice of $\mathbf{x}(s)$ as the parametric representation simplifies many investigations.

Unit tangent vector: The vector $\mathbf{t}(s) = \underset{h\rightarrow 0}{\lim} \frac{\mathbf{x}(s+h)-\mathbf{x}(s)}{h} = \frac{d\mathbf{x}}{ds} = \mathbf{\dot{x}}(s)$ is called the unit tangent vector to the curve $C$ at point $\mathbf{x}(s)$.

The straight line passing through a point $P$ of $C$ in the direction of the corresponding unit tangent vector is called the tangent to the curve at $P$. The tangent can be represented in the form $\mathbf{y}(u) = \mathbf{x} + u\mathbf{t}$. Thus, $\mathbf{y}(0)$ is the point of contact between $C$ and the tangent.

The totality of all vectors bound at a point $P$ of $C$ which are orthogonal to the corresponding unit tangent vector lie in a plane. This plane is called the normal plane to $C$ at $P$.