Test 3,Math 3105 Name:_________________________
Fall 2001, Dr.Howard
Please show all work and justify all answers on the blank pages provided.
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1.Find ∫F6dRwhere F=(y+2)i+xj,along the C:x=sint,y= −cost, 0 ≤t≤π
2.
2.Give an example (or asketch)of aregion in the plane that is not adomain.
3.Give an example (or asketch)of adomain in the plane that is connected,but is not simply
connected.
4.One of the following vector fields is conservative and one is not.Determine which one is
conservative and find an associated potential function.
F=ex+yi+exy j
G=(2x+y)i+(zcosyz +x)j+(ycosyz)k
5.Determine the element of surface area dS for the surface parameterized by
x=(5+2cosv)cosu
y=(5+2cosv)sinu
z=2sinv
6.Let Sbe the closed cylinder of radius 3with axis along the z-axis,top at z=15, and bottom at
z=0. Calculate ∫∫SzdS.
7.Verify the Divergence Theorem for the vector field F=(y−x)i+(y−z)j+(x−y)kover the unit
cube D=[0,1]×[0,1]×[0,1].
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