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Math 6
Spring 2007
Final Review
Directions: all problems by hand unless indicated otherwise by the presence of
an (M) at the beginning of the problem.
1. What conditions have to apply to all the integral theorems for their conclusions to
hold?
2. State each of the vector integral theorems in appropriate notation
3. Describe what each of the above theorems says in words.
4. Show specifically how each of the theorems is related to the fundamental theorem
of calculus.
5. Write in cylindrical coordinates
a. the equation of the cylinder x 2  y 2  r 2
b. The equation of the cone z  x 2  y 2
6. What specifically does each of the following refer to and in what kind of integrals
is it used?
a. ds
b. dS
c. dS
d. dA
Write the area element dA in
i. rectangular coordinates
ii. polar coordinates
e. n
f. T
g. dV
Write the volume element dV in
i. rectangular coordinates
ii. cylindrical coordinates
iii. spherical coordinates
h. dt
i. dz dy dx
j. rdrd
k. rdzdrd
7. Evaluate by hand (sometimes, one of the integral theorems will help)
a.  ( x 2  y 3 )dx  3 xy 2 dy , where C is the portion of y = x2 between x = 0
C
and x = 2, followed by the line segments between the origin and (2,0) and
then the segment from (2,0) to (2,4)
b.  ( x2  y)dx  y 2 dy counterclockwise along the unit circle
c.

S
3z dS , where S is the portion of the plane 2x + y + z = 2 lying in the
first octant
d. Use spherical coordinates to find

S
(3x 2  3 y 2  3z 2 )dS , where S is the
sphere x 2  y 2  z 2  4
e. Find the flux of the vector field F  x, y, 0  over the portion of the
paraboloid z  x 2  y 2 below z = 4.
f. Let Q be the part of the paraboloid z  4  x 2  y 2 above the xy-plane.
Using cylindrical coordinates, find the flux of the vector field
F  x 3 , y 3 , z 3 through the solid whose upper surface is Q and whose
bottom is the intersection of Q in the xy-plane.
1
g.
 
1 x 2
1  1 x 2

2 x2  y 2
x2  y 2
3
( x 2  y 2 ) 2 dzdydx
8. Prove that the curl of a conservative vector field is zero.
9. Prove that  (F  G )  G (  F)  F (  G )
10. Write down a vector field R that has positive divergence and negative curl.
11. Write down the triple integral that represents the volume of any solid.
12. Write the double integral that represents the area of any closed, planar curve.
13. (M) Find the surface area of the part of the paraboloid z  1  x 2  y 2 that lies
below the plane z = 5.
14. Maximize y-x subject to the constraint that the solution lie on the unit circle.
15. If the temperature on an elliptical plate is given by T  x 2  2 x  y 2 and the plate
has a defining equation of x 2  4 y 2  24 , find the maximum and minimum
temperatures on the plate as well as their locations.
16. The plane x + y + z = 12 intersects the paraboloid z  x 2  y 2 in an ellipse. Find
the point on the ellipse that’s closest to the origin.
17. Locate and classify all critical points for the surface f ( x, y)  2 x 2  y 3  2 xy
2 g
u 2
19. Find the direction of maximum change of f at the given point as well as the value
of the change:
a. f ( x, y)  x 2  y 3 at (2,1)
b. f ( x, y )  x cos 3 y at (2,0)
20. For a real gas, van der Waal’s equation states that

n2 a 
P


 V  nb   nRT
V2 

P = pressure, V = volume, T = temperature, n = moles of gas, R is the universal gas
P
T
constant, and a and b are constants. Compute and interpret
and
V
P
18. If g (u, v)  f ( x(u, v), y (u, v)) , find