  Many modified gravity theories are under consideration in cosmology as the
source of the accelerated expansion of the universe and linear perturbation
theory, valid on the largest scales, has been examined in many of these models.
However, smaller non-linear scales offer a richer phenomenology with which to
constrain modified gravity theories. Here, we consider the Hu-Sawicki form of
$f(R)$ gravity and apply the post-Friedmann approach to derive the leading
order equations for non-linear scales, i.e. the equations valid in the
Newtonian-like regime. We reproduce the standard equations for the scalar
field, gravitational slip and the modified Poisson equation in a coherent
framework. In addition, we derive the equation for the leading order correction
to the Newtonian regime, the vector potential. We measure this vector potential
from $f(R)$ N-body simulations at redshift zero and one, for two values of the
$f_{R_0}$ parameter. We find that the vector potential at redshift zero in
$f(R)$ gravity can be close to 50\% larger than in GR on small scales for
$|f_{R_0}|=1.289\times10^{-5}$, although this is less for larger scales,
earlier times and smaller values of the $f_{R_0}$ parameter. Similarly to in
GR, the small amplitude of this vector potential suggests that the Newtonian
approximation is highly accurate for $f(R)$ gravity, and also that the
non-linear cosmological behaviour of $f(R)$ gravity can be completely described
by just the scalar potentials and the $f(R)$ field.
