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MVC Quiz #4 NAME: Instructions: SHOW ALL WORK !! No Calculator allowed. 1.(7 pts) Suppose F x, y, z 2 xy z i x 2 y j x 1 k . Determine whether or not F is a conservative vector field (i.e. is there a scalar function f such that F f ?). If it is, find the function f for which F f . If not, explain clearly. 2. (5 pts) Let f be a scalar field and F a vector field. Indicate which of the following are scalar fields (S), vector fields (V), or meaningless (M). a. div f b. f c. F d. div( f ) e. curl( f ) f. (div F) g. curl(curl F) h. div(div F) i. ( f ) j. div(curl( f )) k. curl(div( f )) MVC 1 3. (8 pts) Find the flow line for the vector field F ( x, y, z ) , 2, z 3 with x(0) (1,5, 2) . x 4. (5 pts) Assuming that f : F f div( F ) f F div f2 f MVC 3 and F : 3 3 are appropriately differentiable, show: 5. (8 pts) Let x (t ) 2cos t , 2sin t , t ,0 t 8 . a. Find the arc-length parameter s (t ) . b. Find the unit tangent T , the unit normal N , and the curvature . c. Find the tangential and normal components of the acceleration. MVC 6. (3 pt’s) Find an integral expression for the length of the curve in which the cylinders b x z 1 and y z 1 . Your answer should be of the form 2 2 2 2 f (t )dt . a MVC