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Page 1 of 6
Name ________________________
Calculus 3
Test #04: Vector Calculus
Assigned: Thursday 5/01/2008
Due: Monday 5/05/2008
You must show your work to get full credit.
Score: __ / 130
1. ( __ / 20 ) Compute the work done by the force field F 
y
,
x
x y
x  y2
(1, 0, 1) along the helix of radius 1 that completes 1 turn over a height of 1.
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2
2
2
, 0 from (1, 0, 0) to
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Page 2 of 6
2. Consider the line integral
Name ________________________
 y
2
 2 x  dx  x 2 dy where C is the square from (0, 0) to (1, 0) to (1, 1) to
C
(0, 1) to (0, 0) as shown below.
a. ( __ / 8 ) Evaluate the line integral directly.
b. ( __ / 7 ) Evaluate the line integral using Green’s Theorem.
c. ( __ / 3 ) If this integral represents the line integral of a vector field along C, state the vector
field being integrated.
d. ( __ / 2 ) State whether or not the vector field is conservative and why.
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Page 3 of 6
Name ________________________
3. Consider the vector field F  sin  x  , 2 y 2 , z
a. ( __ / 7 ) Compute the divergence of this vector field.
b. ( __ / 11 ) Compute the curl of this vector field.
c. ( __ / 2 ) State whether or not this vector field is irrotational and why.
4. ( __ / 10 ) If A is a constant vector and r  x, y, z , prove that   A  r   2A
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Page 4 of 6
Name ________________________
5. ( __ / 20 ) Evaluate the surface integral
 zdS , where S is the hemisphere z  
9  x2  y 2 .
S
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Page 5 of 6
Name ________________________
6. ( __ / 20 ) Verify the divergence theorem by computing both integrals for F  x 2 , 2 y,  x 2 , where Q is
the region bounded by x  2 y  z  4 and the coordinate planes.
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Page 6 of 6
Name ________________________
7. ( __ / 20 ) Verify Stokes’s theorem by computing both integrals for F  2 x, z 2  x, xz 2 , where S is the
portion of z  1  x 2  y 2 above the xy plane.
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