Definition:Divergence Operator

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Physical Interpretation

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

The divergence of $\mathbf V$ at a point $P$ is the total flux away from $P$ per unit volume.

It is a scalar field.

Geometrical Representation

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:

\(\ds \operatorname {div} \mathbf V\) \(:=\) \(\ds \nabla \cdot \mathbf V\)
\(\ds \) \(=\) \(\ds \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}\)


$\nabla$ 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.

Real Cartesian Space

Let $\map {\R^n} {x_1, x_2, \ldots, x_n}$ denote the real Cartesian space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

Let $\mathbf f = \tuple {\map {f_1} {\mathbf x}, \map {f_2} {\mathbf x}, \ldots, \map {f_n} {\mathbf x} }: \R^n \to \R^n$ be a vector-valued function on $\R^n$.

Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$.

The divergence of $\mathbf f$ is defined as:

\(\ds \operatorname {div} \mathbf f\) \(:=\) \(\ds \nabla \cdot \mathbf f\)
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial u_k} } \cdot \mathbf f\) Definition of Del Operator
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {\partial f_k} {\partial x_k}\)

Riemannian Manifold

Definition 1

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

Let $\mathbf X : \map {\CC^\infty} M \to \map {\CC^\infty} M$ be a smooth vector field.

The divergence of $\mathbf X$ is defined as:

\(\ds \operatorname {div} \mathbf X\) \(:=\) \(\ds \nabla \cdot \mathbf X\)
\(\ds \) \(=\) \(\ds \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X}\)


$\star_g$ is the Hodge star operator of $\struct {M, g}$
$\d_{\d R}$ is de Rham differential.

Definition 2

Let $\struct {M, g}$ be an oriented Riemannian manifold.

Let $\rd V_g$ be the volume form of $\struct {M, g}$.

Let $X$ be a smooth vector field on $M$.

Then the divergence of $X$, denoted by $\operatorname {div} X$, is defined by:

$\paren {\operatorname {div} X} \rd V_g := \map d {X \mathop \lrcorner \rd V_g}$

where $\lrcorner$ denotes the interior multiplication, and $\map \rd {X \mathop \lrcorner \rd V_g}$ is the exterior derivative of $X \mathop \lrcorner \rd V_g$.

Also known as

The divergence of a vector field $\mathbf V$ is usually vocalised div $\mathbf V$.

Also see

  • Results about divergence can be found here.

Historical Note

During the course of development of vector analysis, various notations for the divergence operator were introduced, as follows:

Symbol Used by
$\nabla \cdot$ or $\operatorname {div}$ Josiah Willard Gibbs and Edwin Bidwell Wilson
$\operatorname {div}$ Oliver Heaviside
Max Abraham
Vladimir Sergeyevitch Ignatowski
Hendrik Antoon Lorentz
Cesare Burali-Forti and Roberto Marcolongo