MVC Chapter 6 Quiz NAME:
I nstructions: SHOW ALL WORK !! Calculator allowed, but all set up must be shown in order to gat full credit. Explain how and when you use calculator.
1. a. Determine a scalar potential for vector field F
 x
 xy
2 i y 2
 x
2 
1 j y 3
. f x
 x
 y
2 xy
2
 y x
2 x ( , )

2 x y
2
2 x
2
 
2
, f y
  x
2 
1
 
1 y
3 y
3
( x
2 
1)
  x
2 y
2
2

1
2 y
2

 
2 x y
2
2
 x
2

2 2
1 y
2

C b. Find the work done by from (0,1) to (1,1).
F in moving a particle along the parabolic path y 1 x x
2

C
F d s

C
( )
 d s
 f (1,1)
 f (0,1)
3 1
2 2
1

C c. Can the Fundamental Theorem of line integrals be used to calculate x
 xy
2 y
2 dx
 x
2 
1 dy y
3 for a path from (1,1) to (1, 1) . Explain briefly .
NO. Any path from (1,1) to (1, 1) crosses the x-axes and hence will contain a point with y = 0.
Therefore the vector field F will not be conservative on an open set containing the path.
MVC