We saw in the thread https://www.physicsforums.com/showthread.php?t=238464" that arc length that is usually defined by taking an arbitrary parametrisation of the curve as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]l(\gamma)=\int_{0}^{1} {|\dot\gamma(t)|} dt[/tex]

can be defined also by avoiding parametrization, introducing the notion of thedistanceof points as

[tex]d(x,y) = \sup\{|a(y)-a(x)| : a \in C(M), \Vert{\mathrm{grad} a\Vert _\infty \leq 1\}[/tex]

where

[tex] \Vert{\mathrm{grad} a\Vert _\infty = sup\{\mathrm{grad} a|_x: x \in M\}[/tex]

(see equation 3.5 on page 34 of http://ncg.mimuw.edu.pl/index.php?option=com_content&task=view&id=148&Itemid=98", thanks gel for finding it).

The defnition of the integral of an 1-form over a curve is also defined usually by taking a parametrization of the curve:

[tex]\int_\gamma \omega =\int_{0}^{1} \omega(\dot\gamma(t)) dt[/tex]

I wondered if we can find a definition of this integral also by avoiding the parametrization.

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# Defining the integral of 1-forms without parametrization

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