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midterm 2005 ans

midterm 2005 ans - Math102 Midterm-2005 Problem 1(a(a b c...

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Math102 Midterm-2005 Problem 1 (a) ( a × b ) × c = ( a · c ) b - ( b · c ) a , i.e. μ = a · c , λ = - b · c and β = 0. The resulting vector of ( a × b ) × c is a linear combination of the vectors a and b , hence it lies on the plane containing the vectors a and b . (b) (i) ∇ · r = 3. (ii) 0 (iii) 2 a . Problem 2 (a) r is a conical helix wound around the cone z = 2 π - p x 2 + y 2 starting at the vertex (0 , 0 , 2 π ) and completing one revolution to end up at (2 π, 0 , 0). The projection curve is z = ( y - 2 π ) sin y . -5 0 5 x -5 0 5 y 0 2 4 6 z (b) (i) r ( τ ) = cos πτ i + sin πτ j , 0 6 τ 6 1 (ii) r ( τ ) = - cos(2 πτ ) i + sin(2 πτ ) j , 0 6 τ 6 0 . 5. Problem 3 s = 10 . 036 a Problem 4 (b) ds = s dt 2 + ( ρ ( t )) 2 dt 2 + ( ρ ( t ) sin φ ( t )) 2 dt 2 dt . (c) L = 4 5. Problem 5 (a) f ( x, y ) is continuous at (0 , 0). (b) f x = 4 x 2 y x 2 + y 2 - 2 y ( y 2 - x 2 ) 2 ( x 2 + y 2 ) 2 f y = - 4 xy 2 x 2 + y 2 + 2 x ( x 2 - y 2 ) 2 ( x 2 + y 2 ) 2 – 1 –
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