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andrzej1167
science forum beginner

Joined: 16 Jan 2006
Posts: 30

Posted: Sat Jul 08, 2006 5:20 pm    Post subject: I am looking for information about nonlinear differential forms and its applications

In the book written by David Bachman
"A Geometric Approach to Differential Forms"
http://arxiv.org/PS_cache/math/pdf/0306/0306194.pdf
I found the following definition of line integral

$\int\limits_{\gamma}f(x,y)\sqrt{dx^2+dy^2}$

where

$ds=\sqrt{dx^2+dy^2}$

is nonlinear differential form.
There is also the following definition

$\int\limits_{S}f(x,y,z)\sqrt{(dy\wedge dz)^2+(dx\wedge dz)^2+(dx\wedge dy)^2}$

of surface integral where

$dS=\sqrt{(dy\wedge dz)^2+(dx\wedge dz)^2+(dx\wedge dy)^2}$

is also nonlinear differential form.
(I know that the statment "definition of integral" is not appropriate
here but probably everybody knows what I mean by these formulas).

--------------------------------------------\\
Does anybody know other references to the theory
or applications of nonlinear differential forms?
--------------------------------------------\\

Let us consider the following differential equation

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

Is this equation equivalent to the following nonlinear differential
form?

$f(x,y)dx^2-dy^2=0$

I know that it is possible to rewrite that equation as\\
$(y'-\sqrt{f(x,y)}) (y'+\sqrt{f(x,y)})=0$ \\
and then get two linear differential form \\
$\sqrt{f(x,y)}dx-dy=0, \sqrt{f(x,y)}dx+dy=0$\\
but I am interested in the theory of nonlinear differential forms.
Not particularly in this example but in some general theory which
deal with nonlinear differential forms and its relations to
differential equations.

----------------------\\
Does anybody know some references to differential equations which
are expressed using nonlinear differential forms?
e.g. $f(x,y)dx^2-dy^2=0$\\
----------------------\\

If I understand the concept correctly the nonlinear differential forms
on some manifold $M$
are the nonlinear functions
$\omega:T_p(M) \ni v_p \rightarrow \omega(v_p)\in R$
where $T_p(M)$ is tangent space to the manifold $M$ in the point $p\in M$ e.g. $\omega=\sqrt{dx^2+dy^2}$
then \\
$\omega(v_p)=\sqrt{dx(v_p)^2+dy(v_p)^2}=\sqrt{ (v_p^x)^2+ (v_p^y)^2}$
\\
In general that function may have the form \\
$\omega(v_p)= \omega(dx^1(v_p),...,dx^n(v_p))$\\
where $dx^1,...,dx^n$ are basis form in cotangent space $T^*_p(M)$.

----------------------------------------\\
When integral from such nonlinear forms have any sense?
e.g.\\
$\int \sqrt{dx^2+dy^2}$ \\
on some curve have well defined sense but what about\\
$\int \sqrt{dx^2+dy^2+dx}$\\
or\\
$\int \sqrt{dx^2+dy^2+dx+5}$\\
----------------------------------------

I will be very grateful for any comments.

Regards,

Andrzej

P.S.
If anybody is close to heart attack

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