  Second-order two-scale expansions, a unified proof for the regularity of the
correctors based on the translation invariant and a lemma for extracting
$O(\epsilon)$ from the remainder term are presented for the second order
nonlinear elliptic equation with rapidly oscillating coefficients.
  If the data are smooth enough, the error of the zero-order (or energy) in
$L^\infty$, first-order in the H\"older norm, (linear periodic
case)second-order's(even first-order's) gradient (or flux) in the maximum
norm,are locally $O(\epsilon)$. It can be used in the parabolic equation.
