hw12s - Math 481 H/wk 12(Solutions Due Friday April 27 1...

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Math 481 H/wk 12 (Solutions), Due Friday, April 27 1. Let X = (3 x - z, yx 2 , xyz + 1) and Y = ( e xz , y + x, z ). (a) Compute D X Y . (b) For c ( t ) = ( t, t 2 , 1 - 3 t ) compute D ˙ c Y . Solution. (a) We have: D X Y = (3 x - z ) D e 1 Y + yx 2 D e 2 Y + ( xyz + 1) D e 3 Y = (3 x - z ) ∂x Y + yx 2 ∂y Y + ( xyz + 1) ∂z Y = (3 x - z )( ze xz , 1 , 0) + yx 2 (0 , 1 , 0) + ( xyz + 1)( xe xz , 0 , 1) = ((3 xz - z 2 + x 2 yz + x ) e xz , 3 x - z + yx 2 , xyz + 1) . (b) Since ˙ c ( t ) = (1 , 2 t, - 3), we have: D ˙ c Y = D (1 , 2 t, - 3) Y = ∂x Y + 2 t ∂y Y - 3 ∂z Y = = ( ze xz , 1 , 0) + 2 t (0 , 1 , 0) - 3( xe xz , 0 , 1) = = ((1 - 3 t ) e t - 3 t 2 , 1 , 0) + (0 , 2 t, 0) - (3 te t - 3 t 2 , 0 , 3) = = ((1 - 6 t ) e t - 3 t 2 , 2 t + 1 , - 3) . 2. Let M = { ( x, y, z ) R 3 : xy + x + 3 zy - z = 0 } be the surface in R 3 with the orientation induced from the manifold-with-boundary N = { ( x, y, z ) R 3 : xy + x +3 zy - z 0 } . We endow M with the Riemannian metric given by the restriction of the standard inner product in R 3 . Note that from the equation of M we have z = yx + x 1 - 3 y for y 6 = 1 / 3. (a) Consider the curves c ( t ) = ( t, 1 , - t ) and γ ( t ) = ( t, 1 + t 2 , - t 3 +2 t 2+3 t 2 ) in R 3 , where t ( - 1 , 1). Verify that c ( t ) and γ ( t ) are curves in M . Then compute ˙ c ˙ c where is the covariant derivative on M . (b) Let α ( t ) = ( α 1 ( t ) , α 2 ( t ) , α 3 ( t )) be a curve in R 3 . Write a system of equations for α 1 , α 2 , α 3 giving the necessary and suffi- cient conditions for α to be a geodesic on M . Note:
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hw12s - Math 481 H/wk 12(Solutions Due Friday April 27 1...

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