- #1

- 319

- 64

(where ##U_1##, ##U_2##, ##U_3## are the natural (rectangular) unit vectors)

express the connection form and dual 1-forms in terms of ##\psi## and the natural coordinates x,y."

For the connection forms I get ##\omega_{12}=-\omega_{21}=d\psi## with the other components zero. For the dual 1-forms I had difficulty. The definition of dual 1-forms states they are the 1-forms ##\theta_i## such that ##\theta_i(v)=v\cdot E_i(p)## for each tangent vector v. I have difficulty applying this definition. One reason is that it is independent of any coordinate system.

I noticed that the given frame field matches the frame field of cylindrical coordinates with the angle ##\psi##. So that was my answer, the dual 1-forms of cylindrical coordinates: ##\theta_1=dr## where ##r=\sqrt{x^2+y^2}## and ##\theta_2=rd\psi##.

So my question is, is my answer correct? Can I just pick a coordinate system and use it to compute the dual 1-forms?