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# HW03sol - Dierential Geometry 1 Homework 03 1 Determine the...

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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 ξ and η defined on R 2 by ξ = x 2 ∂y , η = y 2 ∂x . Determine which (if any) of the following vector fields are complete: ξ, η, [ ξ, η ] , ξ + η. 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 0 , y (0) = y 0 , z (0) = z 0 . (a) Thoroughly pedestrian. Repeated differentiation: 4 z = 2( x + y ) = y + z + z + x = 2 z + ( x + y ) = 2 z + 2 z or 2 z - 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 0 ; setting t = 0 in 2 z yields 2 a - b = x 0 + y 0 . Conclusion: z = 1 3 ( x 0 + y 0 + z 0 ) e t + (2 z 0 - x 0 - y 0 ) e - t/ 2 1

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with symmetric expressions for x and y . (b) With a twist. Notice that ( x + y + z ) = x + y + z whence s = s 0 e t where s = x + y + z . Now s 0 e t = x + y + z = 1 2 ( y + z ) + 1 2 ( z + x ) + z = z + 2 z is a first-order linear equation for z which may be solved by the usual tech-
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HW03sol - Dierential Geometry 1 Homework 03 1 Determine the...

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