= Negative-divergence obstruction to invariant tori
{title2=$\nabla\cdot F<0$}
A smooth embedded invariant two-torus in three-dimensional Euclidean <phase space> bounds a solid <torus> of positive volume. The <vector field> is tangent to its boundary, so its normal flux is zero. If the <divergence> is strictly negative throughout the enclosed volume, the <divergence theorem> gives a strictly negative flux, a contradiction. Thus a globally negative-divergence three-dimensional <vector field> has no such quasiperiodic <invariant torus>. This argument is stronger than merely noting contraction of transported phase volume; an <attractor>'s own zero volume is not a contradiction.
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