- #1

- 200

- 1

I'd like to calculate the work done by the gravitational force. I know the work is defined by the integration of a 1-form:

[tex]L=\int_\gamma \omega[/tex]

where

[tex]\omega=F_xdx+F_ydy+F_zdz[/tex]

This works fine in cartesian coordinates and I know how to integrate it, but what if I want to use spherical coordinates?

Then I'd have:

[tex]\omega=F_rdr+F_{\theta}d{\theta}+F_{\phi}{d\phi}=F_rdr[/tex]

Suppose [tex]\gamma[/tex] is a curve defined in spherical coordinates (i.e. [tex]\vec\gamma=R(t)\hat r+\Theta(t)\hat\theta+\Phi(t)\hat\phi[/tex]),

how do I integrate the 1-form along [tex]\gamma[/tex]?