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What is the relation between differential equations and differential forms?
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

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

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 "Injection-Info:" 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

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

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)dx-dy=0

which is actually a differential form.
-----------------------------------
Let us consider nonlinear equation

(y')^2=f(x,y)

the expression

f(x,y)dx^2-dy^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

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

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

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

"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

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

Solution of differential equation

y'=f(x,y)

Is equivalent to the solution of the equation

f(x,y)dx-dy=0

which is actually a differential form.
-----------------------------------
Let us consider nonlinear equation

(y')^2=f(x,y)

the expression

f(x,y)dx^2-dy^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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