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?
$f(x,y)dx^2dy^2=0$
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)}dxdy=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. $f(x,y)dx^2dy^2=0$\\
\\
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
after reading my post
then please simply ignore it:). 
