midterm 2005 sol

midterm 2005 sol - HKUST MATH 102 Midterm Examination...

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HKUST MATH 102 Midterm Examination Multivariable and Vector Calculus 21 Dec 2005 Answer ALL 8 questions Time allowed – 180 minutes Directions – This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit. Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet. Student Name: Student Number: Tutorial Session: Question No. Marks 1 /20 2 /20 3 /20 4 /20 5 /20 6 /20 7 /20 8 /20 Total /160 – 1 –
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Problem 1 (20 points) Your Score: (a) Assume a , b and c are three dimensional vectors and if ( a × b ) × c = λ a + μ b + β c . Use suffix notation to find λ , μ and β in terms of the vectors a , b and c . Can you say something about the direction of the vector ( a × b ) × c . (b) Let a be a constant vector and r = ( x, y, z ), use suffix notation to evaluate (i) ∇ · r , (ii) ∇ · ( a × r ), (iii) ∇ × ( a × r ). Solution: (a) [( a × b ) × c ] i = ijk ( a × b ) j c k = ijk jpq a p b q c k = jki jpq a p b q c k = ( δ kp δ iq - δ kq δ ip ) a p b q c k = a k b i c k - a i b k c k i.e. ( 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 . a b c b a (a b) c x x x Plane P is normal to the plane P (b) (i) ∇ · r = i r i = 3. (ii) ∇ · ( a × r ) = i ( a × r ) i = i ε ijk a j r k = ε ijk a j i r k = ε ijk a j δ ik = ε iji a j = 0 (iii) [ ∇ × ( a × r )] i = ε ijk j ( a × r ) k = ε ijk j ε kpq a p r q = ε kij ε kpq a p δ jq = ε kij ε kpj a p = ( δ ip δ jj - δ ij δ jp ) a p = 3 a i - a i = 2 a i ∇ × ( a × r ) = 2 a . – 2 – Problem 2 (20 points) Your Score: (a) Sketch and describe the parametric curve C r = t cos t i + t sin t j + (2 π - t ) k , 0 6 t 6 2 π. Show the direction of increasing t . Find the project curve C onto the yz -plane. (b) Find a change of parameter t = g ( τ ) for the semicircle r ( t ) = cos t i + sin t j , 0 6 t 6 π such that (i) the semicircle is traced counterclockwise as τ varies over the interval [0 , 1], (ii) the semicircle is traced clockwise as τ varies over the interval [0 , 0 . 5]. Solution: (a) x = t cos t y = t sin t z = 2 π - t x 2 + y 2 = t 2 t = p x 2 + y 2 since t > 0 . i.e. z = 2 π - p x 2 + y 2 . 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 com- pleting one revolution to end up at (2 π, 0 , 0).
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