MVC
Quiz #4
Instructions: SHOW ALL WORK !!
NAME:
#1(8 pts) Let F ( x, y, z )   6 x  2 y  z  iˆ   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. 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 ))
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2

#3(8 pts) Find the flow line for the vector field F ( x, y, z )   3 x, y 3 ,  with x(0)  (5, 2, 2) .
z

#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 (8 pts) Let x (t )   et cos t , et sin t  ,0  t  4 .
a. Find the arclength.
b. Find the unit tangent T , the unit normal N , and the curvature  at time t.
MVC
#6(3 pts) Prove the following theorem: (You can get a point for a picture that illustrates the
theorem).
Theorem (The Generalized Mean Value Theorem) If x :[a, b]  2 be a smooth path in
g (c) g (b)  g (a)
.
x (t )  ( f (t ), g (t )) . Then there exists a value c  (a, b) such that

f (c) f (b)  f (a)
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2
with