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Let $X=\sum_ja_j(x)\frac{\partial }{\partial x_j}$ be a $BV$ vector field in an open subset of $\mathbb R^n.$ Alberti's theorem says that $$ DX_s=(\frac{\partial a_j }{\partial x_k})_{1\le j,k\le n}=S \otimes \eta, \quad \text{$S(x)$ tangent vector at $x$, $\eta(x)$ cotangent vector at $x$} $$ i.e for $T$ tangent at $x$, $ (DX)_s(x) T= \langle \eta(x), T\rangle S(x). $ Intuitively, it means essentially that, near each point, you can find a coordinate system such that each coefficient $a_j$ of your vector field is in fact a function $$ a_j(x', x_n) \text{ such that}\ \frac{\partial a_j}{\partial x'} \in L^1,\quad \frac{\partial a_j}{\partial x_n} \text{is a Radon measure}, $$ where $x'$ is $n-1$ dimensional and $x_n$ has just one dimension.