Math 251 Week in Review Sections 14.5-14.6
1. Find a potential function or show the vector field is not conservative.
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a) F ( x , y , z )  x i  e sin z j  e cos z k
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y
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y
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b) F ( x , y , z )  xz i  xy j  yz k
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c) F ( x , y , z )  ( 2 xy z  y ) i  (3 x y z  2 yz  3 xy ) j  ( x y  y ) k
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2. F  xy
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3
2
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i  yz
2
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3
2 2
2
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2 3
2
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j  zx k Does there exist a vector field G for which
2
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F  curl (G ) ? Hint: What is div( F )?
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3. a) Show that any vector field of the form f ( x ) i  g ( y ) j  h ( z ) k is irrotational.
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b) Show any vector field of the form f ( y , z ) i  g ( x , z ) j  h ( x , y ) k is incompressible.
4. Parameterize the.upper half of the ellipsoid 4 x  9 y  z  36 :
2
2
2
a) using x and y as parameters. b) using a polar form on 2x and 3y. c) using a spherical form with φ and
θ as parameters.
5. Find the surface area of the paraboloid z  x  y
2
6. Find the surface area of the cylinder x  z
2
2
2
0  z  9.
 a 2 that lies inside y 2  z 2  a 2 .
7. Find an equation of the tangent plane, at the point (1,0,1), to the surface parameterized as
x (u , v )  u 2
y (u , v )  u  v 2
z  v2 .