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 ))
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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 
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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.
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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
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