Definition:Divergence Operator/Real Cartesian Space

Definition

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

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \dfrac {\partial f_k} {\partial x_k}\)

Cartesian $3$-Space

In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:

Let $R$ be a region of Cartesian $3$-space $\R^3$.

Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.

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

\(\ds \operatorname {div} \mathbf V\)

\(:=\)

\(\ds \nabla \cdot \mathbf V\)

\(\ds \)

\(=\)

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

Definition of Dot Product

Also see

• Results about the divergence operator can be found here.