s10wk04 - Math 23 B. Dodson Week 4: 13.3 arc length,...

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Math 23 B. Dodson Week 4: 13.3 arc length, curvature 13.4 velocity, acceleration; Week 4 Homework: 13.3 curvature Problem 13.3.16 Use formula (9) to find the curvature of ± r ( t ) = < t 2 , 2 t, ln t > . Solution: We start with part (a), find the unit tangent ± T and principal unit normal ± N. We compute ± r ± = < 2 t, 2 , 1 t > and | ± r ± | 2 = 4 t 2 + 4 + 1 t 2 = (2 t + 1 t ) 2 , so | ± r ± | = 2 t + 1 t (since t > 0 , is used for ln t ).
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2 . We also simplify 1 | ± r ± | = t 2 t 2 + 1 , then ± T = 1 | ± r ± | ± r ± = t 2 t 2 + 1 < 2 t, 2 , 1 t > . Using the product rule, ± T ± = ± t 2 t 2 + 1 ± < 2 t, 2 , 1 t > + t 2 t 2 + 1 ( < 2 t, 2 , 1 t > ) ± = ± (2 t 2 + 1) - t (4 t ) (2 t 2 + 1) 2 < 2 t, 2 , 1 t > + t 2 t 2 + 1 ± < 2 , 0 , - 1 t 2 > = ± 1 (2 t 2 + 1) 2 [( - 2 t 2 + 1) < 2 t, 2 , 1 t > + t (2 t 2 + 1) < 2 , 0 , - 1 t 2 > ] = ± 1 (2 t 2 + 1) 2 < 4 t, - 4 t 2 + 2 , - 4 t > . To find the principal unit normal direction, we drop the (positive) scalar factor of 2 (2 t 2 + 1) 2 , and take the direction of < 2 t, - 2 t 2 + 1 , - 2
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This note was uploaded on 03/26/2010 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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s10wk04 - Math 23 B. Dodson Week 4: 13.3 arc length,...

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