= Total variation under opposite smooth flows
{title2=$\operatorname{TV}(w\circ\Phi_t)+\operatorname{TV}(w\circ\Phi_{-t})-2\operatorname{TV}(w)=O(t^2)$}
For the <local flow> of a smooth compactly supported <vector field> and any scalar <BV space> function $w$, the <change of variables formula> and <polar decomposition of a vector measure> give
$$
\operatorname{TV}(w\circ\Phi_t)=\int|\operatorname{cof}(D\Phi_{-t})\sigma_w|\,d|Dw|.
$$
The two <cofactor matrices> are $I\pm tB+O(t^2)$ uniformly. On $|\sigma_w|=1$, the linear terms in their Euclidean <norm> expansions cancel. Integration bounds the remainder by $Ct^2|Dw|(\Omega)$. The identity includes all components of the <decomposition of a BV derivative>; it does not require smoothness of $w$ or its jump interfaces.
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