UBC_MATH_317_2011_101 - December 2011 Mathematics 317 Name...

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December 2011 Mathematics 317 Name Page 2 of 12 pages Marks [18] 1. Short answers. Answer each question below. Read and think carefully. For this question only, no explanation or justification is needed, and no credit will be given for an incorrect answer. (There are no typos in this question. If something looks incorrect, you should say so in your answer.) (a) (3 marks) True or false? If r ( t ) is the position at time t of an object moving in R 3 , and r ( t ) is twice differentiable, then | r 00 ( t ) | is the tangential component of its acceleration. (b) (3 marks) Let r ( t ) is a smooth curve in R 3 with unit tangent, normal and binormal vectors T ( t ) , N ( t ), B ( t ). Which two of these vectors span the plane normal to the curve at r ( t )? (c) (3 marks) True or false? If F = P i + Q j + R k is a vector field on R 3 such that P, Q, R have continuous first order derivatives, and if curl F = 0 everywhere on R 3 , then F is conservative. Continued on page 3
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December 2011 Mathematics 317 Name Page 3 of 12 pages (d) (3 marks) True or false? If F = P i + Q j + R k is a vector field on R 3 such that P, Q, R have continuous second order derivatives, then curl(div F ) = 0 . (e) (3 marks) True or false? If F is a vector field on R 3 such that | F ( x, y, z ) | = 1 for all x, y, z , and if S is the sphere x 2 + y 2 + z 2 = 1, then RR S F · d S = 4 π .
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