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MVC Quiz #4 Instructions: SHOW ALL WORK !! Calculator Allowed – 55 minutes NAME: #1(5 pts) Let F ( x, y, z ) 3x 2 y 2 y iˆ x 3 2 x ˆj ( x)kˆ a) Calculate the curl of F . b) 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. #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. div(div F) d. (div F) e. curl( f ) MVC c. div((curl F)) 2 Find the flow line, x x(t ) for the vector field F ( x, y, z ) 3 x,3, with z x(0) (5,5, 2) . #3(8 pts) #4 (5 pts) Assuming that f : div( fF ) f F f div( F ) MVC 3 and F : 3 3 are appropriately differentiable, show: #5(5 pts) The function F ( x, y) 3x3 y 2 9 x 4 y has critical points at (1, 2) and (1, 2) . That is, F (1, 2) = F (1, 2) (0, 0) . Determine whether each of these critical points corresponds to a local maximum, a local minimum, or a saddle point. #6(5 pts) Find all local extrema for the function g ( x, y, z) x y 2 z ln xyz MVC #7 (2 pts) Find the arc-length of the cardiod r () 1 cos(), 0 2. . MVC