Definition:Divergence Operator/Integral Form

Definition
Let $R$ be a region of space embedded in a Cartesian coordinate frame.

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

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


 * $\ds \operatorname {div} \mathbf V := \lim_{\delta \tau \mathop \to 0} \dfrac {\int_S \mathbf V \cdot \d S} {\delta \tau}$

where:
 * $S$ is the surface of a volume element $\delta \tau$ containing $A$
 * $\cdot$ denotes the dot product
 * $\int_S$ denotes the surface integral over $S$.

Also see

 * Equivalence of Definitions of Divergence


 * Justification for Geometrical Representation of Divergence Operator