  We derive rigorous one- and two-dimensional mean-field equations for cigar-
and pancake-shaped dipolar Bose-Einstein condensates with arbitrary
polarization angle. We show how the dipolar interaction modifies the contact
interaction of the strongly confined atoms. In addition, our equations
introduce a nonlocal potential, which is anisotropic for pancake-shaped
condensates. We propose to observe this anisotropy via measurement of the
condensate aspect ratio. We also derive analytically approximate density
profiles from our equations. Both the numerical solutions of our reduced
mean-field equations and the analytical density profiles agree well with
numerical solutions of the full Gross-Pitaevskii equation while being more
efficient to compute.
