# Divergence

In vector analysis, the **divergence** of a differentiable vector field **F**(**r**) is given by an expression involving the operator nabla (**∇**)—also known as the *del* operator. The definition of nabla and divergence are given by the following equations:

where **e**_{x}, **e**_{y}, **e**_{z} form an orthonormal basis of . The dot stands for a dot product. In the older literature one finds the notation div **F** for **∇**⋅**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 **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

Multiply the left- and right-hand side by an infinitesimal volume element Δ*V* containing the point **r**. Then the left hand side gives the mass leaving Δ*V* minus the mass entering Δ*V* (per unit time). The right-hand becomes equal to which is the rate of decrease in mass. Hence the net flow of mass leaving the the volume Δ*V* is equal to the decrease of mass in Δ*V* (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

Such a vector field **φ**(**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* **∇**⋅**F** = 0, we have

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.