  We consider the inverse problem of determining the coefficients of a general
second-order elliptic operator in two dimensions by measuring the corresponding
Cauchy data on an arbitrary open subset of the boundary. We show that one can
determine the coefficients of the operator up to natural obstructions such as
conformal invariance, gauge transformations and diffeomorphism invariance.
  We use the main result to prove that the curl of the magnetic field and the
electric potential are uniquely determined by measuring partial Cauchy data
associated to the magnetic Schroedinger equation measured on an arbitrary open
subset of the boundary. We also show that any two of the three coefficients of
a second order elliptic operator whose principal part is the Laplacian, are
uniquely determined by their partial Cauchy data.
