  We systematically analyze the effects of the derivatives of the hydrodynamic
field on axial Wigner function that describes the spin polarization vector in
phase space. We have included all possible first-order derivative contributions
that are allowed by symmetry and compute the associated transport functions at
one-loop using the linear response theory. In addition to reproducing known
effects due to the temperature gradient and vorticity, we have identified a
number of potentially significant contributions that are overlooked previously.
In particular, we find that the shear strength, the symmetric and traceless
part of the flow gradient, will induce a quadrupole for spin polarization in
the phase space. Our results, together with hydrodynamic gradients obtained
from hydrodynamic simulations, can be employed as a basis for the
interpretation of the $\Lambda$ (anti-$\Lambda$) spin polarization measurement
in heavy-ion collisions.
