Math 251 Week in Review Sections 14.1-14.4
1. Plot some vectors in each quadrant for each vector field.


a ) F ( x, y ) 


yix j
x2  y2

c ) F ( x , y )   f ( x , y ) f ( x , y )  ln
2. Evaluate


b) F ( x, y )  x i  y j
x2  y2
 f ( x, y ) ds.
C
a)
f ( x, y ) 
x2  y2
b)
f ( x, y )  e x  y
x  t cos t , y  t sin t , 0  t  
C is the triangle with vert ices (0,0), (1,0) and (0,1)
3. Evaluate each line integral.
a )  sin y dy
C : y  x2 , 0  x  
C
b )  x y dx  2 y x dy
C:
y  1  x2
from (1,0 ) to ( 0,1)
C

c)

 F  dr
F  x , xy , z
2
2

C : r (t )  sin t , cos t , t 2
0t  2
C

4. Find the work done if the force F ( x , y )  x , xy is applied once counterclockwise
2
a) around the circle x  y  4 .
2
2
b) around the boundary of the triangle from (0,0) to (2,0) to (0,2) and back to (0,0)
5. Evaluate each line integral.

a)
  f  dr
f ( x, y )  ( x  y )e x
2
 y2
C : x  4 cos t
y  4 sin t
0t  2
C

b)




F ( x , y )   y sin xy i  x cos xy j C is the line segment from (0,0) to (1,  3).
 F  dr
C
 
c)
 F  dr



F ( x , y )  (  y sin( xy )  y ) i  ( x sin( xy )) j
C
C is the line segment from (0,0) to (1,  3).
6. Find a potential function if possible or show the force function is not conservative.

a)
F ( x , y )  ( ye

b)
F ( x , y )  xye
xy

 4 x y ) i  ( xe
3
x2 y

ix e
j


F ( x , y )  ( y cos x  cos y ) i  (sin x  x sin y ) j

d)
4

2 xy

c)

x ) j
xy


F ( x , y )  ( e 2 x  x sin y ) i  x 2 cos y j
 

7. Show F is independent of path and evaluate
 F  dr .
C

a)
C:
F ( x , y , z )  2 xy z i  3 x y z
t, t 2 , t3
3 4
2
2 4


j  4x y z k
2 3 3
0t2

b)



F ( x , y )  ( 2 y  12 x y  3 x ) i  ( 4 xy  9 x y  4 y ) j
2
3 3
2
4
2
3
C is the line segment from (0,0) to (2,3)

1
c) F ( x , y )   2 i  j C is any path, not crossing the y - axis, from
x
x
(1,1) to (2, 6). Does the Fundamenta l Theorem apply for the line segment from
(-2,3) to (2,5)?
y

8. Evaluate



 F  dr
F  ye x
2
 y2

i  xe x
2
 y2

j
C is the line segment
C
from (0,0) to (1,0) followed by the quarter of the unit circle from (1,0) to (0,1)
followed by the line segment from (0,1) to (0,0).
9. Evaluate
 (3 x
y  xy ) dx  4 x 3 y 3 dy C is the boundary of the quarter circle
2 4
C
of radius 1 from (1,0) to (0,1). Note that C is not closed but the integrals on the
line segments (0,1) to (0,0) and (0,0) to (1,0) are 0.
10. Evaluate using Green’s Theorem.

a)

 F  dr

F  ye
x2  y2

i  xe
x2  y2

j
C is the boundary of the sector of the
C
0  
unit circle
b)
(y
2

4
,
0  r  1.
 arctan x ) dx  (3 x  sin y ) dy
C is the curve y  x 2 from (-2,4)
C
to (2,4) followed by the line segment from (2,4) to (-2,4).

c)

 F  dr



F  xy i  y arcsin x j
2
C is the boundary of the quarter of the
C
unit circle from (1,0) to (0,1).
Note : C is not closed.