  In this paper we show how a second order scalar uniformly elliptic equation
on divergence form with measurable coefficients and Dirichlet boundary
conditions can be transformed into a first order elliptic system with
half-Dirichlet boundary condition. This first order system involves Hodge-Dirac
operators and can be seen as a natural generalization of the Beltrami equation
in the plane and we develop a theory for this equation, extending results from
the plane to higher dimension. The reduction to a first order system applies
both to linear as well as quasilinear second order equations and we believe
this to be of independent interest. Using the first order system, we give a new
representation formula of the solution of the Dirichlet problem both on simply
and finitely connected domains. This representation formula involves only
singular integral operators of convolution type and Neumann series there of,
for which classical Calder\'on-Zygmund theory is applicable. Moreover, no use
is made of any fundamental solution or Green's function beside fundamental
solutions of constant coefficient operators. Remarkably, this representation
formula applies also for solutions of the fully non-linear first order system.
We hope that the representation formula could be used for numerically solving
the equations. Using these tools we give a new short proof of Meyers' higher
integrability theorem. Furthermore, we show that the solutions of the first
order system are H\"older continuous with the same H\"older coefficient as the
solutions of the second order equations. Finally, factorization identities and
representation formulas for the higher dimensional Beurling-Ahlfors operator
are proven.
