Divergence

In vector analysis, the divergence of a differentiable vector field F(r) is given by  an  expression involving the operator nabla (&nabla;)&mdash;also known as the del operator. The definition of nabla and divergence are given by the following equations:
 * $$ \begin{align} \boldsymbol{\nabla}\cdot \mathbf{F}(\mathbf{r}) &\equiv \left(\mathbf{e}_x \frac{\partial}{\partial x}+\mathbf{e}_y \frac{\partial}{\partial y}+\mathbf{e}_z \frac{\partial}{\partial z}\right) \cdot \left(\mathbf{e}_x F_x(\mathbf{r})+\mathbf{e}_y F_y(\mathbf{r})+\mathbf{e}_z F_z(\mathbf{r}) \right) \\ &=\frac{\partial F_x(\mathbf{r})}{\partial x}+\frac{\partial F_y(\mathbf{r})}{\partial y}+\frac{\partial F_z(\mathbf{r})}{\partial z}, \end{align} $$

where ex, ey, ez form an orthonormal basis of $$\scriptstyle \mathbb{R}^3$$. The dot stands for a dot product. In the older literature one finds the notation div F for &nabla;&sdot;F.

Physical meaning
The physical meaning of divergence is given by the continuity equation. Consider a compressible fluid (gas or liquid) that is in flow. Let &phi;(r,t)  be its flux (mass per unit time passing through a unit surface)  and let &rho;(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 &Delta;V containing the point r. Then the left hand side gives the mass leaving &Delta;V minus the mass entering &Delta;V (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 &Delta;V is equal to the decrease of mass in &Delta;V (both per unit time).

If the fluid is incompressible, i.e., the mass density &rho; 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 &phi;(r,t) is called divergence-free, solenoidal, transverse, or circuital.

Note
From the Helmholtz decomposition of a vector field it follows that a divergence-free vector field can be written as the curl of another vector field, i.e., provided the longitudinal component &nabla;&sdot;F = 0, we have
 * $$ \mathbf{F} = \boldsymbol{\nabla}\times \mathbf{A}, $$

where A is sometimes referred to as the vector potential. A very well-known example of a divergence-free field is a magnetic field B, which is divergence-free  by virtue of one of Maxwell's equations. The vector field A is then the magnetic vector potential.