Vector Fields
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Transcript of Vector Fields
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Vector Fields
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1. Computer ||F|| and sketch several representative vectors in the vector field
(Similar to p.1067 #7-16)
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2. Find the conservative vector field for the potential function by finding its gradient
(Similar to p.1067 #21-29)Hint: F(x, y) = fxi + fyj
23 4),( yxyxf
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3. Find the conservative vector field for the potential function by finding its gradient
(Similar to p.1067 #21-29)Hint: F(x, y) = fxi + fyj
323),,( zyx eezyxf
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4. Verify that the vector field is conservative(Similar to p.1067 #31-34)
Hint: F(x, y) = Mi + Njconservative if:
x
N
y
M
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5. Determine whether the vector field is conservative.
(Similar to p.1067 #35-38)Hint: F(x, y) = Mi + Nj
conservative if:
x
N
y
M
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6. Determine whether the vector field is conservative.
(Similar to p.1067 #35-38)Hint: F(x, y) = Mi + Nj
conservative if:
x
N
y
M
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7. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field(Similar to p.1067 #39-48)
Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect
to the variable to get the potential function
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8. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field(Similar to p.1067 #39-48)
Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect
to the variable to get the potential function
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9. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field(Similar to p.1067 #39-48)
Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect
to the variable to get the potential function
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10. Find curl F for the vector field at the given point(Similar to p.1067 #49-52)
ky
M
x
Nj
z
M
xi
z
N
PNMzyx
P
y
P z)y,F(x, curl
kji
z)y,F(x, curl
z) y, F(x, z)y,F(x, curl
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Test for Conservative Vector Field in Space
Suppose that M, N, and P have continuous first partial derivatives in an open sphere Q in space. The vector field given by F(x, y, z) = Mi + Nj + Pk is conservative if and only ifCurl F(x, y, z) = 0That is, F is conservative if and only if:
y
M
x
N
z
M
x
P
z
N
y
P
and , ,
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11. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field(Similar to p.1068 #57-62)Hint: F(x, y) = fxi + fyj + fzk
after you get fx, fy, and fz, take the integral with respect to the variable to get the potential function
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Definition of Divergence of a Vector Field
The divergence of F(x, y) = Mi + Nj is:
y
N
x
MyxF
),( y) F(x, div
The divergence of F(x, y, z) = Mi + Nj + Pk is:
z
P
y
N
x
MzyxF
),,( z)y, F(x, div
If div F = 0, then F is said to be divergence free
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12. Find the divergence of the vector field F(Similar to p.1068 #63-66)
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13. Find the divergence of the vector field F(Similar to p.1068 #63-66)
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14. Find the divergence of the vector field Fat the given point
(Similar to p.1068 #63-66)