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Lee Rudolph science forum Guru
Joined: 28 Apr 2005
Posts: 566

Posted: Fri Jul 07, 2006 5:26 pm Post subject:
Re: What is the relation between differential equations and differential forms?



israel@math.ubc.ca (Robert Israel) writes:
Quote:  In article <e8liut$4n1$1@panix2.panix.com>,
Lee Rudolph <lrudolph@panix.com> wrote:
"andrzej" <question2006a@hotmail.com> writes:
I will be gratefully for any references
which are related to relation between
nonlinear differential equations and differential forms.
Will you be, indeed? Under your previous name,
"andrzej" <jedrek_2004@hotmail.com>, you seemed
ungrateful indeed for all attempts to inform your
willful ignorance; why should we believe you are
not engaged in the same kind of trolling once again?
Andrzej is a very common name in Poland.

And is the "Google Groups" posting account
0VIq4g0AAAALIdXtcD0twijLWH3jqv1L (as found
in the "InjectionInfo:" headers both of this
latest post and of "andrzej"'s posts throughout
the thread "How to define a vector without coordinate
system?") also very common in Poland?
Quote:  I don't
think you should assume this is the same one.

Reverend Bayes made me do it.
Lee Rudolph 

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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Fri Jul 07, 2006 4:39 pm Post subject:
Re: What is the relation between differential equations and differential forms?



In article <1152272324.755744.288030@m73g2000cwd.googlegroups.com>,
andrzej <question2006a@hotmail.com> wrote:
Quote:  Solution of differential equation
y'=f(x,y)
Is equivalent to the solution of the equation
f(x,y)dxdy=0
which is actually a differential form.

Let us consider nonlinear equation
(y')^2=f(x,y)
the expression
f(x,y)dx^2dy^2=0
isn't a differential form.
Then what is it?

Try the two differential forms sqrt(f(x,y)) dx  dy and
sqrt(f(x,y)) dx  dy (when f(x,y) > 0)
Quote:  
Let us consider an equation
Sin(y')+y'=0
Is it possible to create equivalent differential form in this case?

This DE is equivalent to y' = 0 (if you're working over the reals).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada 

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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Fri Jul 07, 2006 4:32 pm Post subject:
Re: What is the relation between differential equations and differential forms?



In article <e8liut$4n1$1@panix2.panix.com>,
Lee Rudolph <lrudolph@panix.com> wrote:
Quote:  "andrzej" <question2006a@hotmail.com> writes:
I will be gratefully for any references
which are related to relation between
nonlinear differential equations and differential forms.
Will you be, indeed? Under your previous name,
"andrzej" <jedrek_2004@hotmail.com>, you seemed
ungrateful indeed for all attempts to inform your
willful ignorance; why should we believe you are
not engaged in the same kind of trolling once again?

Andrzej is a very common name in Poland. I don't
think you should assume this is the same one.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada 

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Lee Rudolph science forum Guru
Joined: 28 Apr 2005
Posts: 566

Posted: Fri Jul 07, 2006 12:10 pm Post subject:
Re: What is the relation between differential equations and differential forms?



"andrzej" <question2006a@hotmail.com> writes:
Quote:  I will be gratefully for any references
which are related to relation between
nonlinear differential equations and differential forms.

Will you be, indeed? Under your previous name,
"andrzej" <jedrek_2004@hotmail.com>, you seemed
ungrateful indeed for all attempts to inform your
willful ignorance; why should we believe you are
not engaged in the same kind of trolling once again?
Lee Rudolph 

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andrzej1167 science forum beginner
Joined: 16 Jan 2006
Posts: 30

Posted: Fri Jul 07, 2006 11:38 am Post subject:
What is the relation between differential equations and differential forms?



Solution of differential equation
y'=f(x,y)
Is equivalent to the solution of the equation
f(x,y)dxdy=0
which is actually a differential form.

Let us consider nonlinear equation
(y')^2=f(x,y)
the expression
f(x,y)dx^2dy^2=0
isn't a differential form.
Then what is it?

Let us consider an equation
Sin(y')+y'=0
Is it possible to create equivalent differential form in this case?
What about PDE?

I will be gratefully for any references
which are related to relation between
nonlinear differential equations and differential forms.
Regards,
Andrzej 

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