  We consider the problem of recovering the coefficient \sigma(x) of the
elliptic equation \grad \cdot(\sigma \grad u)=0 in a body from measurements of
the Cauchy data on possibly very small subsets of its surface. We give a
constructive proof of a uniqueness result by Kenig, Sj\"ostrand, and Uhlmann.
We construct a uniquely specified family of solutions such that their traces on
the boundary can be calculated by solving an integral equation which involves
only the given partial Cauchy data. The construction entails a new family of
Green's functions for the Laplacian, and corresponding single layer potentials,
which may be of independent interest.
