  This paper concerns the reconstruction of an anisotropic conductivity tensor
$\gamma$ from internal current densities of the form $J = \gamma\nabla u$,
where $u$ solves a second-order elliptic equation $\nabla\cdot(\gamma\nabla u)
= 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum
number of such functionals equal to $n + 2$, where $n$ is the spatial
dimension, is sufficient to guarantee a local reconstruction. We show that
$\gamma$ can be uniquely reconstructed with a loss of one derivative compared
to errors in the measurement of $J$. In the special case where $\gamma$ is
scalar, it can be reconstructed with no loss of derivatives. We provide a
precise statement of what components may be reconstructed with a loss of zero
or one derivatives.
