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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 )
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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 )
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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
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#7 (2 pts) Find the arc-length of the cardiod r ()  1  cos(), 0    2. .
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