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Unformatted text preview: Homework 8 Stephen Taylor June 4, 2005 Pages 67- 68 : 5. Let γ ( t ) = exp(( i- 1) t- 1 ) for 0 < t ≤ 1 and γ (0) = 0. Show that γ is a rectifiable path and find V ( y ). Give a rough sketch of the trace of γ . We apply Proposition (1.3) noting that γ is a piecewise smooth function to find V ( γ ) = lim a → | i- 1 | Z 1 a 1 t 2 e- t- 1 dt = lim a → √ 2 e- t- 1 1 a = √ 2 e- 1 7. Show that γ : [0 , 1] → C , defined by γ ( t ) = t + it sin 1 t for t 6 = 0 and γ (0) = 0, is a path but is not rectifiable. Sketch this path. To show gamma is a path, it suffices to show continuity on the prescribed interval. So we consider lim t → t + it sin 1 t = 0 since the first term goes to zero by continuity and the second since the sine factor is a bounded function multiplied by zero. To show this curve is not rectifiable, we represent it in R 2 as y = x sin 1 /x . Differentiating we find y = sin 1 x- 1 x cos 1 x Setting the above equal to zero and rearranging we find the equation tan(...
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
- Three '09