Divergence

The divergence of a differentiable vector field F(r) is given by the following expression,

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {F} (\mathbf {r} )&\equiv \left(\mathbf {e} _{x}{\frac {\partial }{\partial x}}+\mathbf {e} _{y}{\frac {\partial }{\partial y}}+\mathbf {e} _{z}{\frac {\partial }{\partial z}}\right)\cdot \left(\mathbf {e} _{x}F_{x}(\mathbf {r} )+\mathbf {e} _{y}F_{y}(\mathbf {r} )+\mathbf {e} _{z}F_{z}(\mathbf {r} )\right)\\&={\frac {\partial F_{x}(\mathbf {r} )}{\partial x}}+{\frac {\partial F_{y}(\mathbf {r} )}{\partial y}}+{\frac {\partial F_{z}(\mathbf {r} )}{\partial z}},\end{aligned}}$

where e_x, e_y, e_z form an orthonormal basis of $\scriptstyle \mathbb {R} ^{3}$. The dot stands for a dot product.

The physical meaning of divergence is given by the continuity equation. Consider a compressible fluid (gas or liquid) that is in flow. Let φ(r,t) be its flux (mass per unit time passing through a unit surface) and let ρ(r,t) be its mass density (amount of mass per unit volume) at the same point r. The flux is a vector field (at any point a vector gives the direction of flow), and the density is a scalar field (function). The continuity equation states that

${\boldsymbol {\nabla }}\cdot {\boldsymbol {\phi }}(\mathbf {r} ,t)=-{\frac {d\rho (\mathbf {r} ,t)}{dt}}.$

Multiply the left- and right-hand side by an infinitesimal volume element ΔV containing the point r. Then the left hand side gives the mass leaving ΔV minus the mass entering ΔV (per unit time). The right-hand becomes equal to $\scriptstyle -\Delta V\,d\rho /dt$ which is the rate of decrease in mass. Hence the net flow of mass leaving the the volume ΔV is equal to the decrease of mass in ΔV (both per unit time).

If the fluid is incompressible, i.e., the mass density ρ is constant, meaning that its time derivative is zero, the flux satisifies

${\boldsymbol {\nabla }}\cdot {\boldsymbol {\phi }}(\mathbf {r} ,t)=0.$

Such a vector field φ(r,t) is called divergence-free, solenoidal, or circuital.