  The dominant radiation mechanism that produces the prompt emission in
gamma-ray bursts (GRBs) remains a major open question. Spectral information
alone has proven insufficient in elucidating its nature. Time-resolved linear
polarization has the potential to distinguish between popular emission
mechanisms, e.g., synchrotron radiation from electrons with a power-law energy
distribution or inverse Compton scattering of soft seed thermal photons, which
can yield the typical GRB spectrum but produce different levels of
polarization. Furthermore, it can be used to learn about the outflow's
composition (i.e. whether it is kinetic-energy-dominated or
Poynting-flux-dominated) and angular structure. For synchrotron emission it is
a powerful probe of the magnetic field geometry. Here we consider synchrotron
emission from a thin ultrarelativistic outflow, with bulk Lorentz factor
$\Gamma(R)=\Gamma_0(R/R_0)^{-m/2}\gg1$, that radiates a Band-function spectrum
in a single (multiple) pulse(s) over a range of radii, $R_0\leq R\leq
R_0+\Delta R$. Pulse profiles and polarization evolution at a given energy are
presented for a coasting ($m=0$) and accelerating ($m=-2/3$) thin spherical
shell and for an off-axis top-hat jet with sharp as well as smooth edges in
emissivity. Four different magnetic field configurations are considered, such
as a locally ordered field coherent over angular scales
$\theta_B\gtrsim1/\Gamma$, a tangled field ($B_\perp$) in the plane transverse
to the radial direction, an ordered field ($B_\parallel$) aligned in the radial
direction, and a globally ordered toroidal field ($B_{\rm tor}$). All field
configurations produce distinct polarization evolution with single (for
$B_\perp$ and $B_\parallel$) and double (for $B_{\rm tor}$) $90^\circ$ changes
in the polarization position angle.
