  Casimir force encodes the structure of the field modes as vacuum fluctuations
and so it is sensitive to the extra dimensions of brane worlds. Now, in flat
spacetimes of arbitrary dimension the two standard approaches to the Casimir
force, Green's function and zeta function, yield the same result, but for brane
world models this was only assumed. In this work we show both approaches yield
the same Casimir force in the case of Universal Extra Dimensions and
Randall-Sundrum scenarios with one and two branes added by p compact
dimensions. Essentially, the details of the mode eigenfunctions that enter the
Casimir force in the Green's function approach get removed due to their
orthogonality relations with a measure involving the right hyper-volume of the
plates and this leaves just the contribution coming from the Zeta function
approach. The present analysis corrects previous results showing a difference
between the two approaches for the single brane Randall-Sundrum; this was due
to an erroneous hyper-volume of the plates introduced by the authors when using
the Green's function. For all the models we discuss here, the resulting Casimir
force can be neatly expressed in terms of two four dimensional Casimir force
contributions: one for the massless mode and the other for a tower of massive
modes associated with the extra dimensions.
