  We present an exact expression for the $1/f$ contribution to the noise of the
CMB temperature and polarization maps for a survey in which the scan pattern is
isotropic. The result for polarization applies likewise to surveys with and
without a rotating half-wave plate. A representative range of survey parameters
is explored and implications for the design and optimization of future surveys
are discussed. These results are most directly applicable to space-based
surveys, which afford considerable freedom in the choice of the scan pattern on
the celestial sphere. We discuss the applicability of the methods developed
here to analyzing past experiments and present some conclusions pertinent to
the design of future experiments. The techniques developed here do not require
that the excess low frequency noise have exactly the $1/f$ shape and readily
generalize to other functional forms for the detector noise power spectrum. In
the case of weakly anisotropic scanning patterns the techniques in this paper
can be used to find a preconditioner for solving the map making equation
efficiently using the conjugate gradient method.
