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I am a bit confused about how I can use covector fields on a differentiable manifold.

John M. Lee writes that they can be integrated in a coordinate independent way so I thought that the covector fields could give me a coordinate independent way of calculating distance over a manifold.

Lets say we are working in R^3. This means that if I have a curve [tex]\gamma: I \rightarrow \mathbb{R}^3[/tex] I can measure how far it stretches in the y-direction by doing the integral,

[tex] \int_\gamma dy .[/tex]

If we change coordinates my covector field, [tex]\omega = dy[/tex] gets pullbacked to [tex]\omega' = dy/dy' dy'[/tex] and we get,

[tex] \int_\gamma \frac{dy}{dy'} dy' .[/tex]

It seems coordinate independent in this sense but what if we would have started with the coordinates dy' form the beginning?

Then we would have arrived at:

[tex] \int_\gamma dy' .[/tex]

Which gives another value right?

What have I missed in this subject? :/

Thanks so much,

All the best!

/ Kontilera

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# Covector fields - did I get them wrong?

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