010.104-2007-2-mid - 2(10 27 1:00-3:00 100 1(15 pts Let f(x...

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2 간고 (10 27 1:00-3:00 ) : : . ( 100 ) 1. (15 pts) Let f ( x, y, z ) = xyz . Compute the following. (a) div( i + j + f k ) (b) curl ( f f ) (c) 2 f 2. Let F ( x, y ) = F 1 ( x, y ) i + F 2 ( x, y ) j be a vector field in R 2 satisfying F ( x, y ) = 1 and d ( t ) be a flow line for F ( x, y ), that is, d ( t ) = F ( d ( t )). (a) (5 pts) Find the length of d ( t ) for a t b . (b) (10 pts) Show that the curvature κ ( t ) of d ( t ) is given by κ ( t ) = p [ F 1 ( d ( t )) · F ( d ( t ))] 2 + [ F 2 ( d ( t )) · F ( d ( t ))] 2 . (Hint: The curvature κ ( t ) of a path c ( t ) in R 2 is defined by κ ( t ) = 1 c ( t ) d dt ( c ( t ) c ( t ) ) . ) 3. (10 pts) Calculate Z 0 1 Z π arccos y e sin x dxdy. 4. (15 pts) Find the volume of the cylinder ( x 2) 2 + y 2 4 bounded above by the sphere x 2 + y 2 + z 2 = 16 and below by the xy plane. 5. (15 pts) Let W be the region in R 3 described by
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