= BV slicing theorem
{c}
{title2=$D_ja=\mathcal L^{n-1}\otimes Da_y$}
For a <BV space> function $a$, almost every restriction $a_y$ to a line parallel to $e_j$ has bounded variation on the corresponding domain slice. Directional derivative variation disintegrates as $|D_ja|=\mathcal L^{n-1}\otimes|Da_y|$. Slice jump values agree with the appropriate oriented <BV traces on a hypersurface> for almost every crossing of a rectifiable jump surface. Integrating slice jump sums produces a surface integral with projection factor $|\nu\cdot e_j|$. Uniform essential bounds on the slices are not asserted for an unbounded function.
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