mat293_exam_2006_questions - University of Toronto Faculty...

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Page 1 of 3 University of Toronto Faculty of Applied Science and Engineering FINAL EXAMINATION 9:10-10:40 am, October 23, 2006 SECOND YEAR -- ENGINEERING SCIENCE Program 5 MAT293F VECTOR CALCULUS Examiner: J. W. Davis _________________________________________________________ Instructions: (1) Closed book examination; no calculators, no aids are permitted (2) Write your name and student number in the space provided below. (3) Answer as many questions as you can. Parts of questions may be answered. (4) The questions are NOT assigned equal marks. (5) Boldface quantities represent vectors. (6) Use the overleaf side of pages for additional or preliminary work. (7) Do not separate or remove any pages from this exam booklet. ____________________________________________________________________ Family Name:________________________________________________________ Given Name:________________________________________________________ Student #:________________________________________________________ ________________________________________________________________________________ The following integrals and formulae may be useful. sin cos sin 22 1 8 1 32 4 θθ θ dC =− + Pdx Qdy Q x P y dR F ndS F dV F dr F dS R C V S S C += ⋅=∇ = × ∫∫ ∫∫ ∫∫ ; $ ; vv v v
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Page 2 of 3 1) a) [5 marks] Sketch the region R that gives rise to the following integral.
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mat293_exam_2006_questions - University of Toronto Faculty...

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