Definition:Divergence Operator
Definition
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:
| \(\ds \operatorname {div} \mathbf V\) | \(:=\) | \(\ds \nabla \cdot \mathbf V\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}\) |
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.
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
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:
| \(\ds \operatorname {div} \mathbf X\) | \(:=\) | \(\ds \nabla \cdot \mathbf X\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X}\) |
where:
- $\star_g$ is the Hodge star operator of $\struct {M, g}$
- $\d_{\d R}$ is de Rham differential.
Integral Form
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 known as
The divergence of a vector field $\mathbf V$ is usually vocalised div $\mathbf V$.
Also see
- Results about the divergence operator 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 |