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

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

where

(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:

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

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

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

*distance*of 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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