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441ex1ss - Dr H Khanal Spring 2008 MA 441 Sample...

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Dr. H. Khanal Spring 2008 MA 441 Sample Test #1 (SOLUTION) ! Read each problem carefully before attempting a solution. Show all work to justify your answers. Answer alone carries no credit. Partial credit only for significant progress towards a correct solution. Box or underline final answers. Problems. 1 .[15] Let v = [ y, - x, 0] represents the velocity field of a fluid flow. Is the flow irrotational? Incompressible? Sketch the vector field. Sol. ∇ × v = [0 , 0 , - 2] 6 = 0 , not irrotational. ∇ · v = 0, incompressible. Streamlines are circles. 2 .[16] Given a scalar function f ( x, y, z ) = yz + x 2 ye z and a point P (3 , - 4 , 0), find a. the direction of maximum increase in f ( x, y, z ) at the point P . b. the directional derivative of f ( x, y, z ) at P in the direction a = [1 , - 2 , 2]. Sol. Here f = [2 xye z , z + x 2 e z , y + x 2 ye z ], and | a | = 1 + ( - 2) 2 + 2 2 = 3. Thus, a. the direction of maximum increase of f ( x, y, z ) at the point P : f ( P ) = f (3 , - 4 , 0) = [ - 24 , 9 , - 40] b. the directional derivative: D f a ( P ) = f ( P ) · a | a | = [ - 24 , 9 , - 40] · [ 1 3 , - 2 3 , 2 3 ] = - 122 3 3 .[24] Let z 1 = 1 + i , z 2 = 1 - i , z 3 = - 4 i .
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