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Tutorial5_sol - R n write F = F 1,F 2,F n where F i R n →...

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MATH1211/10-11(1)/Tu5sol Department of Mathematics, The University of Hong Kong MATH1211 Multivariable Calculus (2010–2011, First semester) Solution to Tutorial 5 Note : The following are brief solutions only. You may need to give more details in your solution. 1. x 0 ( t ) = ( t sin t, t cos t, 0); k x 0 ( t ) k = | t | = t (because t 0). Hence T ( t ) = x 0 ( t ) k x 0 ( t ) k = (sin t, cos t, 0) , N = d T /dt k d T /dt k = (cos t, - sin t, 0) k (cos t, - sin t, 0) k = (cos t, - sin t, 0) , B = T × N = (sin t, cos t, 0) × (cos t, - sin t, 0) = (0 , 0 , - 1) . Note that ds dt = k x 0 ( t ) k = t . Hence curvature is κ ( t ) = d T ds = k d T /dt k ds/dt = k (cos t, - sin t, 0) k t = 1 t . Also d B ds = (0 , 0 , 0). Since d B ds = - τ N , the torsion τ is 0. 2. We need to verify that x 0 ( t ) = F ( x ( t )). In fact F ( x ( t )) = (cos t, - sin t, 2) = d dt sin t, d dt cos t, d dt 2 t = d dt x ( t ) = x 0 ( t ) , 3. (a) ∇ · F = ∂x 2 ∂x + ∂y 2 ∂y = 2 x + 2 y . (b) ∇ · F = 0. 4. ∇ × F = fl fl fl fl fl fl fl fl i j k ∂x ∂y ∂z x + y x 2 z - 3 y fl fl fl fl fl fl fl fl = ( - 3 - x 2 ) i + 0 j + (2 xz - 1) k . div (curl F ) = ∂x ( - 3 - x 2 ) + ∂z (2 xz - 1) = - 2 x + 2 x = 0 . Remark : Since F is in C 2 , by Theorem 4.4 on p.218 we know that div (curl F ) must be zero. 5. Suppose vector field F in R n , write
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Unformatted text preview: R n , write F = ( F 1 ,F 2 ,...,F n ) where F i : R n → R . For f F to make sense we must have f : R n → R also. Then ∇ · ( f F ) = ∇ · ( fF 1 ,fF 2 ,...,fF n ) = n X i =1 ∂ ( fF i ) ∂x i = n X i =1 ² ∂f ∂x i F i + f ∂F i ∂x i ¶ = n X i =1 ² ∂f ∂x i F i ¶ + n X i =1 ² f ∂F i ∂x i ¶ = ( ∇ f ) · F + f ( ∇ · F ) = F · ∇ f + f ∇ · F . 6. We calculate f,f ,f 00 ,f 000 as follows: f ( x ) = √ x, f (1) = 1; f ( x ) = 1 2 x-1 / 2 , f (1) = 1 2 ; f 00 ( x ) =-1 4 x-3 / 2 , f 00 (1) =-1 4 ; f 000 ( x ) = 3 8 x-5 / 2 , f 000 (1) = 3 8 . Thus p 3 ( x ) = f (1)+ f (1)( x-1)+ f 00 (1) 2! ( x-1) 2 + f 000 (1) 3! ( x-1) 2 = 1+ 1 2 ( x-1)-1 8 ( x-1) 2 + 1 16 ( x-1) 3 . 1...
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