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s07wk05 - Math 23 B Dodson Week 5 13.3 arc length curvature...

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Math 23 B. Dodson Week 5: 13.3 arc length, curvature 13.4 velocity, acceleration; 14.1 functions of several variables Week 5 Homework: 13.3 curvature (Mon) 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 t >, with length squared 4 t 2 + ( - 2 t 2 + 1) 2 + 4 t 2 = 4 t 2 + (4 t 4 - 4 t 2 + 1) + 4 t 2 = 4 t 4 + 4 t 2 + 1 = (2 t 2 + 1) 2 , so N = 1 2 t 2 +1 < 2 t, - 2 t 2 + 1 , - 2 t > .
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