ࡱ> ;=:` Ubjbj 8b%>>>8>v?lv2?B"BBCX&X Xuuuuuuu$wh?zHv]HYTxXHYHYvBC_vV^V^V^HYPB:CuV^HYuV^V^rhiTV phjC? Pc`{+>Yit\uv0viz[Dzhjzhj XXV^X XSXXXvv]jXXXvHYHYHYHY(~7$~7 Section 10.3: Arc Length and Curvature Practice HW from Stewart Textbook (not to hand in) p. 714 # 1-4 odd, 11-19 Arc Length Suppose we want to calculate the length of vector function r(t ) = f (t) i + g (t) j + h (t) k over some interval  EMBED Equation.3 .  We determine the arc length using the following formula.  Formula for Arc Length For a vector valued function r(t ) = f (t) i + g (t) j + h (t) k, the arc length L of the curve for the closed interval from t = a to t = b is given by the formula  EMBED Equation.3  Example 1: Find the length of the curve  EMBED Equation.3 . . Solution: % Note: The following graph represents a sketch of the vector valued function  EMBED Equation.3   Curvature Suppose we are given the graph of the vector function r(t ) = f (t) i + g (t) j + h (t) k given by the curve C.  Recall that the formula for the unit tangent vector is given by  EMBED Equation.3  The unit tangent vector indicates the direction of the curve. We would like to be able to measure how fast the curve changes, which corresponds to how fast the direction of the unit tangent vector changes. The curvature measures how quickly the curve changes direction at a specific point. It can be shown to be found by the following formula. Formula for Curvature For a vector valued function r(t ) = f (t) i + g (t) j + h (t) k with unit tangent vector T, the curvature  EMBED Equation.3  is given by the formula  EMBED Equation.3  Note: It can be shown that an alternative (and often) easier method for finding the curvature is given by the formula  EMBED Equation.3  Example 2: Find the curvature of the vector function  EMBED Equation.3 . Use your result to find the curvature at the point  EMBED Equation.3  Solution: % Calculating the Unit Normal Vector In describing motion, sometimes it can be an advantage to calculate a vector orthogonal to the unit tangent vector T of the vector function r(t ) = f (t) i + g (t) j + h (t) k.  One method is the use the fact that the T is a unit vector. Then, we know that  EMBED Equation.3  If we use the product rule and differentiate both sides of this equation, we obtain  EMBED Equation.3  or  EMBED Equation.3 . Thus,  EMBED Equation.3  Hence, the derivative of T,  EMBED Equation.3 , is orthogonal to T. Normalizing T will give the principal unit normal vector N at t. Principal Unit Normal Vector N  For the vector valued function r(t ) = f (t) i + g (t) j + h (t) k with unit tangent vector T, the unit normal vector, assuming  EMBED Equation.3 , is given by  EMBED Equation.3  Note: Unless the function r(t) is simple, the unit normal vector N can be very computationally intensive. Maple can be a very convenient tool for simplify the computations. Example 3: Find the unit normal vector N for the vector  EMBED Equation.3 . Use your result to find the unit normal vector at the point  EMBED Equation.3  Solution: To find the unit normal vector N, we must first find the unit tangent vector T. Recall that the formula for the unit tangent vector T is given by  EMBED Equation.3  For  EMBED Equation.3 , we have  EMBED Equation.3  and  EMBED Equation.3  Thus,  EMBED Equation.3  To get the unit normal vector, we use the formula  EMBED Equation.3  Using  EMBED Equation.3 , we see that  EMBED Equation.3  and (continued on next page)  EMBED Equation.3  Then,  EMBED Equation.3  We last want to evaluate  EMBED Equation.3  at the point  EMBED Equation.3 . We need to find the value of t that produces this point. 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Thus,   EMBED Equation.3  The following shows a framed graph of  EMBED Equation.3  (in green) with the unit tangent vector  EMBED Equation.3  (in red) and unit normal vector  EMBED Equation.3  (in blue) displayed at the point  EMBED Equation.3 .  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