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

$\ds \int_S$ denotes the surface integral over $S$.

Also see

• Equivalence of Definitions of Divergence

• Justification for Geometrical Representation of Divergence Operator

• Results about the divergence operator can be found here.

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Appendix $\text B$: Fields and differential operators: $\text B.1$ The Operators Div, Grad and Curl: $(\mathbf B 1)$