the expressions for the potential derivatives

the metric

the inverse of the metric

phi2eta= is derived earlier.

foo110=phi2eta but is separated into and in the summation.

foo111=foo110 but and are substituted by their expressions in the local coordinates.

foo112=

Separate the regular indices a and b for the summation into type-a indexes and 9's.

foo114[ua,ub]= is the same as foo112, but the subscripts are changed. In the below, we substitute 's by their expressions at the projection.

foo115=foo114[aua,aub]

check if foo115 is consistent with ph2bu obtained earlier.

foo116=foo114[9,aua]

foo117=foo114[9,9]

Now obtain the coefficients in terms of for foo115,foo116,foo117.