Rank one property for derivatives of functions with bounded variation

If $u$ is a function of bounded variation from the open set $\Omega \subset R^n$ into $R^m$, then $Du$ is a measure on $\Omega$ which takes values in the space of $m\times n$ matrices, and we denote by $D_Su$ the singular part of this measure (with respect to Lebesgue measure). We prove that that the density of $D_su$ with respect to its variation $|D_su|$ is a function with values in rank-one matrices. More generally, we show that given a singular measure $\mu$, there exists a unit vectorfield $\nu$ such that for every scalar $BV$ function $u$, the density of $Du$ with respect to $\mu$ at $x$ is a multiple of $\nu(x)$ for $\mu$-almost every $x$.