  We show that color-kinematics duality is a manifest property of the equations
of motion governing currents and field strengths. For the nonlinear sigma model
(NLSM), this insight enables an implementation of the double copy at the level
of fields, as well as an explicit construction of the kinematic algebra and
associated kinematic current. As a byproduct, we also derive new formulations
of the special Galileon (SG) and Born-Infeld (BI) theory.
  For Yang-Mills (YM) theory, this same approach reveals a novel structure --
covariant color-kinematics duality -- whose only difference from the
conventional duality is that $1/\Box$ is replaced with covariant $1/D^2$.
Remarkably, this structure implies that YM theory is itself the covariant
double copy of gauged biadjoint scalar (GBAS) theory and an $F^3$ theory of
field strengths encoding a corresponding kinematic algebra and current.
Directly applying the double copy to equations of motion, we derive general
relativity (GR) from the product of Einstein-YM and $F^3$ theory. This exercise
reveals a trivial variant of the classical double copy that recasts any
solution of GR as a solution of YM theory in a curved background.
  Covariant color-kinematics duality also implies a new decomposition of
tree-level amplitudes in YM theory into those of GBAS theory. Using this
representation we derive a closed-form, analytic expression for all BCJ
numerators in YM theory and the NLSM for any number of particles in any
spacetime dimension. By virtue of the double copy, this constitutes an explicit
formula for all tree-level scattering amplitudes in YM, GR, NLSM, SG, and BI.
