Question 2: What does Equation (4) here say? State the theorem used to determine this equation.

Hint 1: Read the subsection "Speed on a Smooth Curve" (p. 783).

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