Divergence

In vector calculus, the divergence of a vector field $\vec{F}$ is defined as:

$$\nabla \cdot \vec{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$

where $\nabla$ is the gradient operator taken over the dot product with the respective $x$, $y$, and $z$ components of the field. The divergence is distributive.

Geometrically, it represents how much the field diverges at a given point, i.e., how much of a vector field is “created” or “expelled” at a point.

If the divergence is zero, $\nabla \cdot \vec{F}=0$, we have a solenoidal or divergence-free field.

Properties

$$\nabla \cdot (\vec{A}+\vec{B})=\nabla \cdot \vec{A}+\nabla \cdot \vec{B}$$ $$\nabla \cdot (\psi \vec{A})=\psi \nabla \cdot \vec{A}+\vec{A}\cdot \nabla \psi$$ $$\nabla \cdot (\vec{A}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{A})-\vec{A}\cdot (\nabla \times \vec{B})$$

Formal statement

For a closed Gaussian surface $S$ enclosing a charge, we let $S$ shrink arbitrarily small to $\Delta S$, and the enclosed volume $V\to \Delta V$. We can express $q_{enc}$ in terms of a volume charge density, ${\rho }_v$.

$$q_{\text{enc}}={\iiint }_v{\rho }_v(x,y,z)dxdydz$$

By Gauss’ law for $\Delta S,\Delta v$:

$${\iint }_{\Delta S}\vec{E}\cdot d\vec{S}=\frac{1}{{ϵ}_0}dq=\frac{{\rho }_v\Delta v}{{ϵ}_0}$$

Then as we let $\Delta S,\Delta v$ approach 0:

$$\underset{\Delta v,\Delta s\to 0}{\lim }\frac{{\iint }_{\Delta S}\vec{E}\cdot d\vec{S}}{\Delta v}=\frac{{\rho }_v}{{ϵ}_0}$$

The formal statement of divergence.

See also

• Curl, the cross product equivalent