  The observation of the polarization emerging from a rotating star at
different phases opens up the possibility to map the magnetic field in the
stellar surface thanks to the well-known Zeeman Doppler Imaging. When the
magnetic field is sufficiently weak, the circular and linear polarization
profiles locally in each point of the star are proportional to the first and
second derivatives of the unperturbed intensity profile, respectively. We show
that the weak-field approximation (for weak lines in the case of linear
polarization) can be generalized to the case of a rotating star including the
Doppler effect and taking into account the integration on the stellar surface.
The Stokes profiles are written as a linear combination of wavelength-dependent
terms expressed as series expansions in terms of Hermite polynomials. These
terms contain the surface integrated magnetic field and velocity components.
The direct numerical evaluation of these quantities is limited to rotation
velocities not larger than 8 times the Doppler width of the local absorption
profiles. Additionally, we demonstrate that, in a rotating star, the circular
polarization flux depends on the derivative of the intensity flux with respect
to the wavelength and also on the profile itself. Likewise, the linear
polarization depends on the profile and on its first and second derivative with
respect to the wavelength. We particularize the general expressions to a
rotating dipole.
