1.
Math 2511: Calc III - Practice Exam 3
State the meaning or definitions of the following terms:
a) Vector field, Conservative vector field, Potential function of a vector field
b) curl and divergence of a vector field F
c)
 dA or  f ( x, y)dA
R
R
d)
 dS
or
e)
 f ( x, y)ds
C
R
 F  dr where F is a two or three dimensional vector field
C
f)
 M ( x, y)dx  N ( x, y)dy
C
g)
 f ( x, y, z )dS
C
 
h)
 F  N dS , where F is a three-dimensional vector field
i)
j)
k)
l)
m)
What does it mean when a “line integral is independent of the path”?
State the Fundamental Theorem of Line Integrals
Please state Green’s Theorem. Make sure to know when it applies, and in what situation it helps
Please state the Divergence Theorem. Make sure to know when it applies, and in what situation it helps
Please state Stokes’ Theorem. Make sure to know when it applies, and in what situation it helps
C
2.
Below are four algebraic vector fields and four sketches of vector fields. Match them.
[A]
[C]
[B]
[D]
(1) F ( x, y)  x, y  , (2) F ( x, y)   y, x  , (3) F ( x, y)  x,1  , (4) F ( x, y)  1, y 
3.
Are the following statements true or false:
a) If the divergence of a vector is zero, the vector field is conservative.
b) If F ( x, y, z ) is a conservative vector field then curl ( F )  0
c)
If a line integral is independent of the path, then
 F  dr  0
for every path C
C
d) If a line integral is independent of the path, then
 F  dr  0
C
for every closed path C
e)
If a vector field is conservative in a disk then
 F  dr  0
for every closed path C inside that disk
C
f)
 dA denotes the surface area of the region R
R
g)
 dS
denotes the volume of the region R
R
h) Can you apply the Fundamental Theorem of line integrals for the function f ( x, y, z)  xy sin(z) cos(x 2  y 2 ) ?
i)
j)
k)
l)
4.
Can you apply the Fundamental Theorem of line integrals for the vector field F ( x, y)  6xy 2  3x 2 ,6x 2 y  3 y 2  7  ?
Can you apply Green’s theorem for a curve C, which is a straight line from (0,0,0) to (1,2,3)?
Can you apply the Divergence theorem for a closed surface S that lies in the xy-plane
Can you apply Stoke’s theorem for a closed surface, i.e. a surface that bounds a solid object in space?
Suppose that F ( x, y, z)  x 3 y 2 z, x 2 z, x 2 y  is some vector field.
a) Find div(F)
b) Find curl(F)
c) Find curl(curl(F))
d) Find div(curl(F))
e) grad., div., and curl of the vector field if appropriate  x 2 , y 2 , z 2 
f)
grad., div., and curl of the vector field if appropriate  cos(y)  y cos(x), sin(x)  x sin( y), xyz 
g) grad., div., and curl of the vector field if appropriate
5.
Decide which of the following vector fields are conservative. If a vector is conservative, find its potential function
a) F ( x, y)  2xy, x 2 
c)
F ( x, y)  e x cos(y), e x sin( y) 
F ( x, y, z )  sin( y),  x cos y,1 
d)
F ( x, y, z)  2xy, x 2  z 2 ,2zy 
e)
F ( x, y)  6xy 2  3x 2 ,6x 2 y  3 y 2  7 
f)
F ( x, y)  2 y 3 sin(2x),3 y 2 (1  cos(2x) 
g)
F ( x, y)  4xy  z,2x 2  6 y,2 z 
h)
F ( x, y)  4xy  z 2 ,2x 2  6 yz,2xz 
b)
6.
f ( x, y, z )  z ln( x 2  y 2 )
Evaluate the following integrals:
a)
 cos(x
2
)dA where R is the triangular region bounded by y = 0, y = x, and x = 1
R
b)
 dS , where S is the portion of the hemisphere
f ( x, y)  25  x 2  y 2 that lies above the circle x 2  y 2  9
R
c)
x
2
 y  3 zds where C is a line segment given by r (t )  t ,2t ,3t  , 0  t  1
C
d)
 F  dr where F ( x, y)  y, x
2
 and C is the curve given by r (t )  4  t,4t  t 2  , 0  t  3
C
e)
 ydx  x
2
dy where C is a parabolic arc given by r (t )  t ,1  t 2  , 1  t  1
C
f)
 ( x  z )dS
where S is the first-octant portion of the cylinder y 2  z 2  9 between x = 0 and x = 4
S
g) Find the flux of the vector field F ( x, y, z)  x, y, z  through the surface given by potion of the paraboloid z  4  x 2  y 2
that lies above the xy-plane. Note that this surface is not closed.
h) In the previous problem, would it be easier if we considered the same surface as before, but joined with the disk of radius 2 in
the xy plane so that the surface would be a closed surface?
7.
For the following line integrals there is a short-cut you can use to simplify your computations (but justify your shortcut by quoting
the appropriate theorem)
a)
 F  dr where F ( x, y, z)  e
x
cos(y),e x sin( y)  and C is the curve r (t )  2 cos(t ), 2 sin(t )  , 0  t  2
C
b)
 2 xyzdx  x
2
zdy  x 2 ydz where C is some smooth curve from (0,0,0) to (1,4,3)
C
c)
 F  dr where F ( x, y)  y
3
 1,3xy 2  1  and C is the upper half of the unit circle, from (1,0) to (-1,0).
C
d)
y
3
dx  ( x 3  3 xy 2 )dy where C is the path from (0,0) to (1,1) along the graph of y  x 3 and from (1,1) to (0,0) along the
C
graph of y  x .
8.
Green’s Theorem
a)
Use Green’s theorem to find
 F  dr where F ( x, y)  y
3
, x 3  3xy 2  and C is the circle with radius 3, oriented counter-
C
clockwise (You may need the double-angle formula for cos somewhere during your computations)
y x
x2 y2
b) Evaluate  dA where R is the ellipse

 1 by using a vector field F ( x, y)   ,  and the boundary C of the
2
2
4
9
R
ellipse R.
9.
Evaluate the following integrals. You can use any theorem that’s appropriate:
c)
 2 xyzdx  x
2
zdy  x 2 ydz where C is a smooth curve from (0,0,0) to (1,4,3)
C
d)
 ydx  2 xdy
where C is the boundary of the square with vertices (0,0), (0,2), (2,0), and (2,2)
C
e)
 xy
2
dx  x 2 ydy , where C is given by r(t )  4 cos(t ),2 sin(t )  , t between 0 and 2 Pi.
C
f)
 xydx  x
2
dy where C is the boundary of the region between the graphs of y  x 2 and y  x .
C
g) If F ( x, y, z)  x, y, z  , find
the plane 2 x  3 y  4z  12
 
  F  N dS
h) If F ( x, y, z)  x 2 , xy, z  , find
and the plane 2 x  3 y  4z  12
i)
If F ( x, y, z)  x, y, z  , find
where S is the surface of the solid region Q bounded by the coordinate planes and
S

 
 F  N dS where S is the surface of the solid region Q bounded by the coordinate planes
S

 F dr where C is the boundary of that portion of the plane 3x  y  2z  12 that lies in the
C
first octant
j)
If F ( x, y, z)  cos(y)  y cos(x),sin(x)  x sin( y), xyz  , find

 F dr where C is the boundary of the surface S given by
C
z  y that lies over the square in the xy-plane with vertices (0,0), (a,0), (a,a), (0, a).
2
10. Prove the following:
a) If F ( x, y, z )  M ( x, y, z ), N ( x, y, z), P( x, y, z )  is any vector field where M , N , P are twice continuously differentiable
then div(curl( F ))  0
b) A function (not a vector field) f ( x, y, z ) is called harmonic if
x 2

2 f
y 2

 2f
z 2
 0 . Show that for any function f ( x, y, z )
1
is harmonic.
f ( x, y, z )
Use Green’s Theorem to prove that integrals of a conservative vector fields over closed curves are zero (if the closed curve
encloses a simply connected region and all conditions of Green’s theorem are satisfied).
the function
c)
2 f