a question about ALE method

hi, I'm a student studying ALE method. In the book of "Nonlinear Finite Elements for Continua and Structures" written by Ted Belytschko el, conservation equations which discribed in the reference configuration and current configuration are given out (see chapter 7). But how about Balance Laws in initial configuration?

In initial configuration, the basis vector is not a function of time and we usually the simply coordinate as the basis(cartesian coordinate or cylindrical coordinate etc.). So when calculating the gradient of something, just give out its partial derivative(except cylindrical coordinate)(for example, in both the Updated and Total Lagrangian FEA method). But in the reference configuration in ALE, the basis is the function of time and the coordinate also should be the curved coordinate(not regular as cylindrical coordinate ), so the gradient of tensor is not too simply to get a partial derivative and the Christoffel symbol maybe included. But in the eqn of moment conservation in Ted Belytschko's book(7.12.9), only partial derivative is mentioned.

I don't know how does this happen. And another question is whether there is any book or article mentioning the ALE form of conservation equations in initial configuration? Is there any kind friend who would like to help me out? I'll appreciate that. Sorry for my poor English.  :)