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The functional $\phi \mapsto -\int u\mathrm{div}\phi$ is bounded in the uniform norm exactly when $u$ belongs to the BV space. Extend it from compactly supported smooth tests to $C_0$, then apply the Riesz-Markov-Kakutani representation theorem to obtain the unique finite vector Radon measure $Du$. Its dual norm is the defining variation supremum. This representation should be distinguished from the deeper rectifiability results used in the decomposition of a BV derivative.

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 giveThe 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 $C{t}^2|Dw|(\Omega )$. The identity includes all components of the decomposition of a BV derivative; it does not require smoothness of $w$ or its jump interfaces.