Exercise 2: Read the subsection "Speed on a Smooth Curve" (p. 783). What does Equation (4) here say? State the theorem used to determine this equation. The rest of Chapter 13 continues on this trend and describes the geometry of curves, but we will return to Chapter 16 where we will briefly use the arc length parameter.

13.3 Arc Length in Space 783 from to to f is Eq. (3) Value from Example 1 = V21. Solving this equation for / gives / = s/ V2. Substituting into the position vector r gives the following are length parametrization for the helix: r((s)) = COS sin Unlike Example 2, the are length parametrization is generally difficult to find analyti- cally for a curve already given in terms of some other parameter t. Fortunately, however, we rarely need an exact formula for s(f) or its inverse r(s). HISTORICAL BIOGRAPHY Speed on a Smooth Curve Josiah Willard Gibbs Since the derivatives beneath the radical in Equation (3) are continuous (the curve is (1839-1903) smooth). the Fundamental Theorem of Calculus tells us that s is a differentiable function www. goo. g1/tsiNim of : with derivative dt ds = [ v()). (4)Equation (4) says that the speed with which a particle moves along its path is the magni- tude of v, consistent with what we know. Although the base point P() plays a role in defining s in Equation (3), it plays no role in Equation (4). The rate at which a moving particle covers distance along its path is inde- pendent of how far away it is from the base point. Notice that ds/di > 0 since, by definition. | | is never zero for a smooth curve. We see once again that s is an increasing function of r. Unit Tangent Vector We already know the velocity vector v = dr/de is tangent to the curve r() and that the vector T = is therefore a unit vector tangent to the (smooth) curve, called the unit tangent vector (Figure 13.15). The unit tangent vector T is a differentiable function of : whenever v is a differentiable function of f. As we will see in Section 13.5, T is one of three unit vectors in FIGURE 13.15 We find the unit tangent a traveling reference frame that is used to describe the motion of objects traveling in three vector T by dividing v by its length | v|. dimensions. EXAMPLE 3 Find the unit tangent vector of the curve r(0) = (1 + 3 cos /)i + (3 sinnj + r-k representing the path of the glider in Example 3, Section 13.2. Solution In that example, we found V = dt Of = -(3 sin n)i + (3 cos nj + 2/k