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Chapter 3 Quiz
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#1. 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 ))
#2. Let r  ( x, y, z ) and let r denote r . Verify that (ln r ) 
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r
.
r2
#3 Determine whether the path x (t )  (et cos t , et (sin t  cos t ), et sin t ) is a flow line for the
vector field F  ( x  z , 2 x, y ) .
#4
Explain why F  x, y, z    2 xy  z  i   x 2  y  j   x  yz  1 k cannot be the gradient of
some scalar field f without mentioning curl. (That is, show that there is no scalar function f
such that F  f ?).
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#5. Calculate the flow line x (t ) of the vector field F ( x, y )   x 2 , 2  that passes through the point
y 

(1,3) at t = 1. (That is, x (1)  (1,3) )
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#6
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.
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