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In article <VXRlh.8205$yC5.3846@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jon Slaughter <Jon_Slaughter@xxxxxxxxxxx> wrote:
>
>"Proginoskes" <CCHeckman@xxxxxxxxx> wrote in message
>news:1167546864.806173.34110@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>>
>> Jon Slaughter wrote:
>>> Is there such a thing?
>>
>> Yes; it's called a potential function.
>>
>> --- Christopher Heckman
>>
>
>Also, not all vector fields are conservative so there is not always a
>potential function involved. I suppose here the inverse
>gradient would be
>singular.
>What I'm takling about is not what the gradient/inverse
>gradient do but the
>*operators* themselfs.
Given a vector field F that has a scalar potential, you can
get that potential, up to a constant, by a line integral:
phi(x) = const + int_{C(x)} F.dr
where C(x) is a rectifiable curve starting at some given
point and ending at x.
F has a scalar potential (on a given domain) iff for every
x in the domain the result does not depend on which curve is
chosen.
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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