  Foreseeing the era of high spatial resolution measurements of the
Sunyaev-Zel'dovich effect (SZE) in clusters of galaxies, we present a prototype
analysis of this sort combined with Chandra X-ray data. It is applied
specifically to RX J1347-1145 at z=0.451, the most X-ray-luminous galaxy
cluster known, for which the highest resolution SZE and X-ray images are
currently available. We demonstrate that the combined analysis yields a unique
probe of complex structures in the intracluster medium, offering determinations
of their temperature, density, and line-of-sight extent. For a subclump in RX
J1347-1145, previously discovered in our SZE map, the temperature inferred
after removing the foreground and background components is well in excess of 20
keV, indicating that the cluster has recently undergone a violent merger.
Excluding the region around this subclump, the SZE signals in submillimeter to
centimeter bands (350, 150, and 21 GHz) are all consistent with those expected
from Chandra X-ray observations. We further present a temperature deprojection
technique based on the SZE and X-ray images, without any knowledge of spatially
resolved X-ray spectroscopy. The methodology presented here will be applicable
to a statistical sample of clusters available in the future SZE surveys.
