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Instructions: Calculator allowed.
SHOW ALL WORK !!
#1(8 pts) Given
Quiz #4
( , , )

3 xy
 yz
2 and ( , , )

NAME:
   
ˆ j

( )
ˆ
, evaluate each of the following if possible. a.
curl( ) b.
div( f F ) c.
div(div( f F ))
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#2(8 pts) Find the flow line, x
 x ( ) for the vector field x (0)
 
.
#3(5 pts) Assuming that
MVC div
 
 

( ) f
2
 f F f :
3  3  3 x
3 t z

2

with
are appropriately differentiable, show:

#4(5 pts) Find all critical points of the function ( , )
 x
2 
2 xy

1
3 y
3 
3 y and determine whether each of these critical points corresponds to a local maximum, a local minimum, or a saddle point.
#5(5 pts) When two resistors having resistances R
1
and R
2 are connected in parallel, the resistance of the combination is given by
R

R
R R
1 2

1
R
2
.
Use differentials to approximate the resistance when resistors having resistances of 2.01 Ohms and
5.99 Ohms are connected in parallel. Show all work without the use of your calculator.
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#6(2 pt’s) Find an integral expression for the length of the curve in which the cylinders x
2  z
2 
1 and y
2  z
2 
1 b
intersect. Your answer should be of the form ( ) a

.
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