  The full sky cosmic microwave background polarization field can be decomposed
into 'electric' (E) and 'magnetic' (B) components that are signatures of
distinct physical processes. We give a general construction that achieves
separation of E and B modes on arbitrary sections of the sky at the expense of
increasing the noise. When E modes are present on all scales the separation of
all of the B signal is no longer possible: there are inevitably ambiguous modes
that cannot be separated. We discuss the practicality of performing E/B
decomposition on large scales with realistic non-symmetric sky-cuts, and show
that separation on large scales is possible by retaining only the well
supported modes. The large scale modes potentially contain a great deal of
useful information, and E/B separation at the level of the map is essential for
clean detection of B without confusion from cosmic variance due to the E
signal. We give simple matrix manipulations for creating pure E and B maps of
the large scale signal for general sky cuts. We demonstrate that the method
works well in a realistic case and give estimates of the performance with data
from the Planck satellite. In the appendix we discuss the simple analytic case
of an azimuthally symmetric cut, and show that exact E/B separation is possible
on an azimuthally symmetric cut with a finite number of non-intersecting
circular cuts around foreground sources.
