Science > Physics > What are the curl vs. conservative filed theorems called?
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Science > Physics |
| User: |
"hetware" |
| Date: |
16 May 2007 03:14:36 PM |
| Object: |
What are the curl vs. conservative filed theorems called? |
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
What are these theorems typically called? He declines to provide a proof,
and there are no references given. Where might I find a shot discussion of
these theorems which include proofs?
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "Dirk Van de moortel" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
16 May 2007 03:57:06 PM |
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"hetware" <massless@nutrino.none> wrote in message news:zrmdnVqAva-z-9bbnZ2dnUVZ_qKqnZ2d@speakeasy.net...
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
What are these theorems typically called? He declines to provide a proof,
and there are no references given. Where might I find a shot discussion of
these theorems which include proofs?
Here
http://en.wikipedia.org/wiki/Conservative_force
and here:
http://en.wikipedia.org/wiki/Conservative_force/Proofs
Dirk Vdm
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| User: "" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
17 May 2007 04:28:32 PM |
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I'm not sure that any such theorem exists, and given your list of
citations, I'm not convenced that you are primarily interested in
physics.
That said, if such such a theorem exists, it is likely "Stokes
Theorem", if I correctly understand your question, which your
citations did not shed any light at all upon.
Mench, you would be well advised to keep you inquiries into physics
here in sci.pysics, and isolating your political/historic views and
post them in an appropriate
newsgroup, since you are risking offending some and recalling horrible
experiences to mind for some others. No thinking human needs to be
reminded of krystalnacht, or equally what happens when "The inmates
take over the asylum"! 'Nuff said.
For substantial introduction to the theory of electromagnetics and
related topice, find a copy of "Principles of Electricity and
Electromagnetism" by Harnwell. This is a McGraw-Hill publication dated
1949, and one which pulls no punches, as some of the watered-down or
dumbed-down popular texts of today do. It will assume that you have a
comprehensive background in calculus and vector analysis, but given
that, it is a wonderful and comprehensive text on electric and
electromagnetics field theory.
Actually, Harnwell's book is so good that it makes Corson & Lorrain's
"Introduction to Electromagnetic Fields and Waves" look like
highschool materials.
Harry C.
On May 16, 4:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
What are these theorems typically called? He declines to provide a proof,
and there are no references given. Where might I find a shot discussion of
these theorems which include proofs?
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "hetware" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
18 May 2007 07:20:40 PM |
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wrote:
I'm not sure that any such theorem exists, and given your list of
citations, I'm not convenced that you are primarily interested in
physics.
That said, if such such a theorem exists, it is likely "Stokes
Theorem", if I correctly understand your question, which your
citations did not shed any light at all upon.
They are more like a corollaries to Stokes's theorem. The reason I wanted a
name for them was so that I could enter the names into my notes.
Mench, you would be well advised to keep you inquiries into physics
here in sci.pysics, and isolating your political/historic views and
post them in an appropriate
newsgroup, since you are risking offending some and recalling horrible
experiences to mind for some others.
Since Germar Rudolf is a chemical physicists who is in prison for reporting
his findings resulting from his scientific research, I have good reason to
call his situation to the attention of this news group.
No thinking human needs to be
reminded of krystalnacht, or equally what happens when "The inmates
take over the asylum"! 'Nuff said.
I suggest you take some time to actually read some of that which you
condemn.
For substantial introduction to the theory of electromagnetics and
related topice, find a copy of "Principles of Electricity and
Electromagnetism" by Harnwell. This is a McGraw-Hill publication dated
1949, and one which pulls no punches, as some of the watered-down or
dumbed-down popular texts of today do. It will assume that you have a
comprehensive background in calculus and vector analysis, but given
that, it is a wonderful and comprehensive text on electric and
electromagnetics field theory.
Actually, Harnwell's book is so good that it makes Corson & Lorrain's
"Introduction to Electromagnetic Fields and Waves" look like
highschool materials.
Harry C.
It looks as though Harnwell may require some effort to obtain. I appreciate
the suggestion, and will keep it in mind. For now, MTW and Feynman
supplemented by Pauli's lectures will be my focus.
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "Eric Gisse" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
16 May 2007 10:59:37 PM |
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On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a proof,
and there are no references given. Where might I find a shot discussion of
these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "hetware" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
25 May 2007 04:55:28 AM |
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Eric Gisse wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot discussion
of these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
As it turns out, Feynman went through the development after all. But he
still didn't give the theorems names.
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "Y.Porat" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
25 May 2007 10:39:38 AM |
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On May 25, 12:55 pm, hetware <massl...@nutrino.none> wrote:
Eric Gisse wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot discussion
of these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
As it turns out, Feynman went through the development after all. But he
still didn't give the theorems names.
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm- Hide quoted text -
- Show quoted text -
-----------
just beware that a curl will not haunt you
while you walk alone at night (:-)
Y.Porat
--------------------
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| User: "Eric Gisse" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
25 May 2007 04:02:43 PM |
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On May 25, 2:55 am, hetware <massl...@nutrino.none> wrote:
Eric Gisse wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot discussion
of these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
As it turns out, Feynman went through the development after all. But he
still didn't give the theorems names.
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
"vector calculus"
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| User: "hetware" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
25 May 2007 05:09:20 PM |
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Eric Gisse wrote:
On May 25, 2:55 am, hetware <massl...@nutrino.none> wrote:
Eric Gisse wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl.
One
says if there is a curl then there is no conservative field. The
other says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot
discussion of these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
As it turns out, Feynman went through the development after all. But he
still didn't give the theorems names.
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
"vector calculus"
Actually, "Vector Integral Calculus" is the title of the chapter.
http://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics
Mary Boas only gives one theorem, but she lists a total of five equivalent
conditions. Still no name, however. Boas's book is clearly something I
have neglected to give proper attention. This is a good book!
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471198269.html
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "hetware" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
18 May 2007 06:51:45 PM |
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Eric Gisse wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot discussion
of these theorems which include proofs?
An introductory mechanics textbook that uses vector calculus.
OK, Symon goes through them, but doesn't elevate the statements to the level
of theorems. Likewise for Menzel, Wrede, McCauley and Joos. My suspicion
is that Goldstein might state them as theorems. All they say is that the
path integral along an arbitrary closed path will vanish if and only if a
force is conservative. Which is basically a definition to my mind.
I say "all that it says", but considering the importance the curl has in
Maxwell's equations, that seems like a fairly important "all".
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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| User: "Y.Porat" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
17 May 2007 08:26:38 AM |
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On May 17, 6:59 am, Eric Gisse <jowr...@gmail.com> wrote:
On May 16, 1:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
If there is a curl, you can't express the force as the gradient of a
scalar function - a potential.
If there isn't, you can.
--------------
i just wonder what is the case once i have
a * Shmerl * and not a curl
i mean a **potential** one ??? a big shmerl ??
TIA
Y.Porat
--------------------------------
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| User: "Mike" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
26 May 2007 01:32:26 AM |
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On May 16, 4:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
What are these theorems typically called? He declines to provide a proof,
and there are no references given. Where might I find a shot discussion of
these theorems which include proofs?
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
I don't think there is any standard name for the theorems. The
facts go like this. Consider the following sequence of operations:
Let me write S for the class of all smooth scalar fields and V for the
class of all smooth vector fields on some open set in R^3. We have
the following sequence of operations:
gradient curl divergence
S ----------> V ----------> V ----------S
The composition of any two consecutive arrows is 0. Thus a vector
field cannot be the gradient of some scalar field unless its curl
vanishes and a vector field cannot be the curl of something unless its
divergence vanishes. When one considers scalar fields and vector
fields without singularities (i.e. defined on ALL of R^3) then we have
a strong converse. In that case for a vector field v, curl v=0 if and
only if there is a scalar field f with grad f = v and also div v = 0
if and only if there is a vector field w with curl w = v. In fancy
jargon, this follows from the fact that the deRham cohomology of R^3
is trivial.
There are important generalizations of these facts that apply to
higher than 3 dimensional spaces. These are important in general
relativity.
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| User: "hetware" |
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| Title: Re: What are the curl vs. conservative filed theorems called? |
26 May 2007 04:59:18 AM |
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Mike wrote:
On May 16, 4:14 pm, hetware <massl...@nutrino.none> wrote:
What are the curl vs. conservative filed theorems called? I have long
been
a ware of this fact, and take it as almost self-evident. Nonetheless,
Feynman mentions two theorems regarding the relationship between a
conservative potential field and the existence of a non-zero curl. One
says if there is a curl then there is no conservative field. The other
says if there is no curl, then there is a conservative field.
What are these theorems typically called? He declines to provide a
proof,
and there are no references given. Where might I find a shot discussion
of these theorems which include proofs?
--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
I don't think there is any standard name for the theorems. The
facts go like this. Consider the following sequence of operations:
Let me write S for the class of all smooth scalar fields and V for the
class of all smooth vector fields on some open set in R^3. We have
the following sequence of operations:
gradient curl divergence
S ----------> V ----------> V ----------S
The composition of any two consecutive arrows is 0. Thus a vector
field cannot be the gradient of some scalar field unless its curl
vanishes and a vector field cannot be the curl of something unless its
divergence vanishes. When one considers scalar fields and vector
fields without singularities (i.e. defined on ALL of R^3) then we have
a strong converse. In that case for a vector field v, curl v=0 if and
only if there is a scalar field f with grad f = v and also div v = 0
if and only if there is a vector field w with curl w = v. In fancy
jargon, this follows from the fact that the deRham cohomology of R^3
is trivial.
One thing about Stokes's theorem which has always left me scratching my head
is this: when discussing fluid flows, when exactly _isn't_ the curl zero?
I can guess that the circulation around a vortex is non-zero, but what
about around a curve that only comes close to the vortex? I believe that
for an idealized fluid, that circulation will vanish. I'm not sure what
happens with a real fluid.
I've also never really been comfortable with the relationship
between "magnetism" and the relativistic effects on the electric field. I
can follow the argument given here:
http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html for example,
but it leaves me a bit unsure.
It clearly does not work to simply consider the wire to be like a subway
tunnel with the electrons as passengers in a train if the train is not
affected by some physical interaction with the tunnel. An unencumbered
train would appear foreshortened, and thus the charge density would be
increased from the point of view of an observer at rest relative to the
tunnel(wire). The wire would appear charged, and would have an effect on
charges at rest. OTOH, if we are driving along outside of the tunnel at
the same speed as the electrons on the train, then the tunnel /is/
foreshortened, and we feel the positive charge excess due to its
electron "holes" (excess protons).
It stands to reason that the "train" is physically stretched out as it is
conducted through the tunnel. That is, from the point of view of the
passenger electrons, the other passengers are further away from them when
the train is in motion. I don't have a problem with that because the
electrons in the wire are continually slamming into things.
From the perspective of the electrons on the train, not only is the train
stretched out due to the rough ride, the tunnel is also foreshortened.
Something is still bothering me about the case in which a charged particle
is at rest relative to a wire with a current running through it. Even if
we argue that the electron train is spread out in its inertial system, from
the perspective of the tunnel (wire) inertial system, the field lines
transverse to the path of travel will be denser than those parallel to the
path of travel. This is due to the way that rigid angles transform. It
therefore seems as though an average neutral charge density would still
produce an electric field. It seems to follow that the electron density in
the wire has to actually _decrease_ when the current flows.
And then there's the whole question of how there can be more electrons in
the stationary wire than in the moving wire. That is, if we simply
integrate the charge density of both electrons and holes, we end up with a
surplus of holes in the case of the wire moving relative to the observer.
That one, I can answer.
There are important generalizations of these facts that apply to
higher than 3 dimensional spaces. These are important in general
relativity.
Any relation to the Bianchi identities?
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
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