  Following Weaver we study generalized differential operators, called (metric)
derivations, and their linear algebraic properties. In particular, for k = 1, 2
we show that measures on k-dimensional Euclidean space that induce rank-k
modules of derivations must be absolutely continuous to Lebesgue measure. An
analogous result holds true for measures concentrated on k-rectifiable sets
with respect to k-dimensional Hausdorff measure.
  Though formulated for Euclidean spaces, these rigidity results also apply to
the metric space setting and specifically, to spaces that support a doubling
measure and a p-Poincar\'e inequality. Using our results for the Euclidean
plane, we prove the 2-dimensional case of a conjecture of Cheeger, which
concerns the non-degeneracy of Lipschitz images of such spaces.
