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Unformatted text preview: Differential Geometry 1 Homework 03 1. Determine the integral curves for the following vector field on R 2 : 1 2 ( y + z ) x + ( z + x ) y + ( x + y ) z . Do so in more than one way! 2. Consider the vector fields and defined on R 2 by = x 2 y , = y 2 x . Determine which (if any) of the following vector fields are complete: , , [ , ] , + . Solutions (1) The integral curves are found by solving the differential equations 2 x = y + z, 2 y = z + x, 2 z = x + y with x (0) = x , y (0) = y , z (0) = z . (a) Thoroughly pedestrian. Repeated differentiation: 4 z 00 = 2( x + y ) = y + z + z + x = 2 z + ( x + y ) = 2 z + 2 z or 2 z 00- z- z = 0 . This differential equation has solutions z = ae t + be- t/ 2 . To find a and b : setting t = 0 in z yields a + b = z ; setting t = 0 in 2 z yields 2 a- b = x + y . Conclusion: z = 1 3 ( x + y + z ) e t + (2 z- x- y ) e- t/ 2 1 with symmetric expressions for x and y ....
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
- Spring '09