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In a Nut Shell:
A
vector field may be conservative or non-conservative. If the curl of the
vector field is zero,
then the vector field is conservative.
A conservative vector
field is said to be irrotational.
A common application
appears in the area of fluid mechanics.
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A vector field, F, can be defined by a
vector valued function at each point (x,y) in a
plane or
by each point (x,y,z) in space such as
F(x, y) = P(x,y) i +
Q(x,y) j
F(x, y, z) =
P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k
where P(x,y), Q(x,y), and R(x,y) are scalar fields (scalar functions)
where
P(x,y,z), Q(x,y,z),
and R(x,yz) are scalar fields (scalar functions)
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Strategy to test for Conservative Vector
Fields: Calculate the curl of
the vector field to
determine if it is
conservative or not. If the curl of
the vector field is zero, then the vector
field is conservative
and there exists a scalar function, a potential function, such that the
gradient of the scalar
function equals the vector function.
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Test for a two-dimensional vector field, F(x,y): curl ( F(x,y)
) = 0
Test for a three-dimensional vector field, F(x,y,z): curl ( F(x,y,z)
) = 0
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For conservative vector fields there exists a
scalar function such that the
vector function equals the gradient of the
scalar function.
F(x,y)
= grad ( f(x,y) )
(two-dimensional case)
F(x,y,z)
= grad ( f(x,y,z) ) (three-dimensional case)
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