# Covector fields - did I get them wrong?

by Kontilera
Tags: covector, fields
 P: 102 Hello! 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 $$\gamma: I \rightarrow \mathbb{R}^3$$ I can measure how far it stretches in the y-direction by doing the integral, $$\int_\gamma dy .$$ If we change coordinates my covector field, $$\omega = dy$$ gets pullbacked to $$\omega' = dy/dy' dy'$$ and we get, $$\int_\gamma \frac{dy}{dy'} dy' .$$ 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: $$\int_\gamma dy' .$$ Which gives another value right? What have I missed in this subject? :/ Thanks so much, All the best! / Kontilera
 P: 350 Howdy, It looks like your question about changing coordinates was answered, but your comment about lengths was unclear, so I thought I'd add something. You won't get lengths from covector fields, rather you'll get net changes. E.g. the integral of dy over your curve returns the net change in the y direction. If you wanted total change in the y direction, you would need to integrate |dy|, which is not strictly speaking a differential form. To measure length, you need a formula for ds. For example to calculate euclidean length there are many possible formulas: $ds = \sqrt{dx^2+dy^2}$ $ds=\sqrt{dx^2+x^2dy^2}$ The specific formula depends on the coordinates you choose. The first corresponds to cartesian coordinates of the plane. The second to polar coords. The formula for the element of arclength is called a riemannian metric. |dy| is like a degenerate metric: $|dy|=\sqrt{0x^2+y^2}$
 P: 102 Thanks! Yeah, net change is a better way to put it. :) Another question that popped up was if it is misleading that we write integration as: $$\int_S f(x,y)\,dxdy \quad ?$$ After all, we are integrating over differential forms, ie: $$\int_S f(x,y)\,dxdy = \int_S f(x,y)\,dx\wedge dy$$ but juxtaposition of dxdy is used for the symmetric product so we should have: $$dxdy = dx \wedge dy$$ Which is not true since the left hand side is symmetric and the right hand side is alternating.