Differential geometry notes 2

by exactnature

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.

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