Pullback - a mathematical puzzler

[carltrembath]

Hi y'all. Carlo T here with a little mathematical puzzler.

Here's the big ol' question, are you folks ready?

let phi:(0, infty)x(0, 2*pi)x(-pi/2, pi/2)->R^3 be defined by
phi(r,alpha,beta)=(r*cos(alpha)*cos(beta), r*sin(alpha)*cos(beta), r*sin(beta)).

Calculate the pullback phi*(xdy)

I've calculated phi*(dx), phi*(dy) and phi*(dz) but am not sure where to go from here.

For example, is there a way of rewriting phi*(xdy) using one of the pullback properties?

Get your noggins round that one brothers.

Peace out, from the C to the T to the remBath!

 

[pat_connell]

The pullback of xdy is given by phi*(xdy)=phi*(dx)*dy + phi*(dy)*dx. We can expand this as phi*(xdy) = r*cos(alpha)*cos(beta)*dy + r*sin(alpha)*cos(beta)*dx.