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Unformatted text preview: Math 410 Homework Solutions February 23, 2009 1 Homework #4 1. Let S be a surface of revolution about the zaxis. It is defined by an equation of the form x 2 + y 2 = f ( z ). Let γ be a geodesic on S and let u ( t ) = ( x ( t ) ,y ( t ) ,z ( t )) be its parametrization by arc length. (a) Derive the EulerLagrange equations for x,y, and z . Solution Consider the following functional F = Z  u ( t )  + λ ( t )( x 2 + y 2 f ( z )) dt For γ to be a geodesic, u must satisfy the Euler equations for this functional, i.e. 2 λx d dt x  u  = 0 2 λy d dt y  u  = 0 λ df dz d dt z  u  = 0 Using the fact that u is parameterized by arc length, we get 2 λx = x 00 2 λy = y 00 λ df dz = z 00 (b) Show that xy yx = c 1 , where c 1 is a constant. Solution Multiplying the first equation by y, second by x, we find yx 00 = xy 00 . But notice d dt ( xy yx ) = xy 00 + x y y x yx 00 = xy 00 yx 00 Therefore, d dt ( xy yx ) = 0, i.e. xy yx = C 1 . 1 (c) Show that the element of arc length in R 3 is given by ds 2 = dr 2 + r 2 dθ 2 + dz 2 . Solution First calculate that dx = cos θdr r sin θdθ dy = sin θdr + r cos θdθ. Then ds 2 = dx 2 + dy 2 + dz 2 = cos 2 θdr 2 + r 2 sin 2 θdθ + sin 2 θdr 2 + r 2 cos 2 θdθ = dr 2 + r 2 dθ 2 + dz 2 . (d) Express the parametrization for the geodesic γ in cylindrical coordi nates u ( t ) = ( r ( t ) ,θ ( t ) ,z ( t )), and find the EulerLagrange equations for r,θ, and z ....
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 Spring '08
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 Math, dt, R2D2

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