Thank you for your response. However, I still am trying to wrap my head around the introduction of a new variable.
After substiuting out dXa,
(dx)^2 - (dX)^2 = (xi,a*xi,a - 1)*dXa*dXa
how does the kronecker delta substitute (dijej = ei) in here to get the next equation?
(dx^2) -...