= Divergence
{title2=$\nabla\cdot$}
{wiki}
For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,
$$
\nabla\cdot F=\partial_iF_i
=\frac{\partial F_1}{\partial x}
+\frac{\partial F_2}{\partial y}
+\frac{\partial F_3}{\partial z}.
$$
For a continuously differentiable <vector field>, the divergence is the limiting outward <flux> per unit volume through a small closed surface. Opposite faces of a small rectangular box almost cancel; the remaining differences in the three components give the sum of <partial derivatives> above. The <divergence theorem> extends this local balance to a whole region.
For $F=(x,y,0)$, the divergence is $1+1+0=2$. In a planar unit square, the right edge contributes outward <flux> $1$ and the top contributes $1$, while the left and bottom contribute zero. The total $2$ equals the integral of the planar divergence over area $1$.
If $F$ is a <velocity field>, positive divergence means local volume expansion, not automatically creation of mass. The <continuity equation> gives $D\rho/Dt=-\rho\,\nabla\cdot F$, where $D/Dt$ is the <material derivative>. For this field, a uniform initial <density> evolves as $\rho(t)=\rho(0)e^{-2t}$ while material volumes grow as $e^{2t}$. Their product stays constant.
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