Definition:Divergence Operator/Geometrical Representation

Definition

Let $\mathbf V$ be a vector field acting over $R$.

The divergence of $\mathbf V$ at a point $A$ in $R$ is defined as:

$\ds \operatorname {div} \mathbf V$

$:=$

$\ds \nabla \cdot \mathbf V$

$\ds $

$=$

$\ds \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}$

where:

$\nabla$ denotes the Del operator

$\cdot$ denotes the dot product

$V_x$, $V_y$ and $V_z$ denote the magnitudes of the components of $\mathbf V$ at $A$ in the directions of the coordinate axes $x$, $y$ and $z$ respectively.