chap7 - Chapter 7: Hints and (partial) Solutions Exercise...

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Chapter 7: Hints and (partial) Solutions Exercise 7.1 (a) Velocity: ˙ α ( t ) = ( α ( t ) , (1 , 2 t )). Acceleration: ¨ α ( t ) = ( α ( t ) , (0 , 2)). Speed: k ˙ α k ( t ) = k ( α ( t ) , (1 , 2 t )) k = 1 + 4 t 2 . (b) Velocity: ˙ α ( t ) = ( α ( t ) , ( - sin t, cos t )). Acceleration: ¨ α ( t ) = - ( α ( t ) , (cos t, sin t )) = - ( α ( t ) ( t )). Speed: k ˙ α k ( t ) = k ( α ( t ) , ( - sin t, cos t )) k = p sin 2 t + cos 2 t = 1. (d) Velocity: ˙ α ( t ) = ( α ( t ) , ( - sin t, cos t, 1)). Acceleration: ¨ α ( t ) = - ( α ( t ) , (cos t, sin t, 0)). Speed: k ˙ α k ( t ) = k ( α ( t ) , ( - sin t, cos t, 1)) k = p sin 2 t + cos 2 t + 1 = 2. Exercise 7.2 Hint : Constant speed implies d dt ( k ˙ α ( t ) k 2 ) = 0. But k ˙ α ( t ) k 2 = ˙ α ( t ) · ˙ α ( t ). Exercise 7.5 Hint : If α is of the from indicated in this problem then argue that α ( t ) S for any t R (Note: a point p S iff p has coordinates ( x,y,z ) with x 2 + y 2 = r 2 ). Now find a vector at α ( t ) that is orthogonal to S at α ( t ). (Note that n p = ( p, ( x,y, 0)) is orthogonal to S at p = ( x,y,z ), why?). Now argue that ¨ α ( t ) = ( α ( t ) , ¨ α ( t )) is a multiple of n α ( t ) , i.e., is orthogonal to S . Conclude α is a geodesic in
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