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| kjetil1001... |
 Posted: Mon Nov 02, 2009 2:12 pm |
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Does anybody know about how to define an integral (of elemnts in a
metric space/manifold)
with values in a metrix space/manifold, that is, given a function f:
\Omega \arrow M where
M is a metric space, an integral
\int_{\Omega} f(x) dx where the integral is an element in M?
Can this be defined? How?
Pointers to papers/books/web sites???
Kjetil halvorsen |
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| Chip Eastham... |
| Posted: Mon Nov 02, 2009 4:29 pm |
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On Nov 2, 7:12 pm, kjetil1001 <kjetil1... at (no spam) gmail.com> wrote:
[quote]Does anybody know about how to define an integral (of elemnts in a
metric space/manifold)
with values in a metrix space/manifold, that is, given a function f:
\Omega \arrow M where
M is a metric space, an integral
\int_{\Omega} f(x) dx where the integral is an element in M?
Can this be defined? How?
Pointers to papers/books/web sites???
Kjetil halvorsen
[/quote]
On smooth (real) manifolds embedded in a
Euclidean space you can define measures
on curves in the manifold, etc. This
allows line integrals (e.g. of real-
valued functions) and various other
kinds of integrals.
However to "integrate" a manifold-valued
function doesn't make sense unless the
manifold has some algebraic structure
that allows scalar multiplication and
(vector) addition of manifold values.
regards, chip |
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| David C. Ullrich... |
| Posted: Tue Nov 03, 2009 7:30 am |
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On Mon, 2 Nov 2009 16:12:39 -0800 (PST), kjetil1001
<kjetil1001 at (no spam) gmail.com> wrote:
[quote]Does anybody know about how to define an integral (of elemnts in a
metric space/manifold)
with values in a metrix space/manifold, that is, given a function f:
\Omega \arrow M where
M is a metric space, an integral
\int_{\Omega} f(x) dx where the integral is an element in M?
Can this be defined? How?
Pointers to papers/books/web sites???
[/quote]
It's hard to imagine a definition of something that would appear
to be the definition of an _integral_ unless the values are in
some sort of vector space at least.
[quote]Kjetil halvorsen
[/quote]
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.) |
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