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.
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