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israel@xxxxxxxxxxx (Robert Israel) writes:
>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.
Better yet (perhaps, for JS's purposes), *any* vectorfield that
is locally a gradient has a *global* potential on some covering
space of the original domain (namely--simplifying without loss
of generality to the case that the original domain D is connected,
and letting x_0 be a fixed basepoint--the covering space whose
points over a point x in D is the set of rectifiable curves C(x)
as above, modulo the equivalence relation "C_0(x) is equivalent
to C_1(x) iff the cycle C_1(x)-C_0(x) is homologous to 0").
Isn't this what Abelian integrals are all about?
Lee Rudolph
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