### Differential geometry notes 2

#### by exactnature

* Definition 9.1:* Let be an allowable parametric representation of an arc with initial point and terminal point . Denote by the length of a broken line of chords whose end points lie on and correspond to the values . Let and . If , then is called

**rectifiable**and is called

**the length**of .

* Theorem 9.1:* Let be an allowable parametric representation of an arc of a curve of class ( times differentiable). Then has length:

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

* Arc length:* The function is called

**the arc length**of .

The arc length may be used as the parameter in a parametric representation of a curve: is an allowable parametric representation. is called **the natural parameter**. The choice of as the parametric representation simplifies many investigations.

* Unit tangent vector:* The vector is called

**the unit tangent vector**to the curve at point .

The straight line passing through a point of in the direction of the corresponding unit tangent vector is called **the tangent** to the curve at . The tangent can be represented in the form . Thus, is the point of contact between and the tangent.

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