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MVC Quiz #4 Instructions: SHOW ALL WORK !! Calculator allowed. 45 minutes 1. z Let F ( x, y) 3x 2 y iˆ y z ˆj ˆj y a) Calculate the curl of F . b) 2. MVC NAME: Is there a scalar function f such that F f ? If so, find f. If no such function exists, explain how you know this to be the case. whether x(t ) (4e2t , e2t , t 2 ) is a flow x F( x, y, z ) x 4 y, ,ln( y ) . Explain work carefully. 2 Determine line for the vector field 2 3. Find the flow line for the vector field F ( x, y, z ) 3 x, y 3 , with x(0) (5, 2, 2) . z 4. If f ( x, y, z ) and F ( x, y, z ) are suitably differentiable, show: div( fF ) f F fdiv( F ) MVC 5. MVC Let x (t ) et cos t , et sin t , et ,0 t 4 . a. Find the arclength parameter s (t ) . b. Find the unit tangent T , the unit normal N , and the curvature . 6. Find an integral expression for the length of the curve in which the cylinders x 2 z 2 1 b and y z 1 . Your answer should be of the form 2 2 f (t )dt . a MVC