Definition:Divergence Operator/Real Cartesian Space

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Definition

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

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

Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \R^n \to \R^n$ 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:

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


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

\(\displaystyle \operatorname {div} \mathbf f\) \(:=\) \(\displaystyle \nabla \cdot \mathbf f\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} }\right) \cdot \mathbf f\) Definition of Del Operator
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\partial f_x} {\partial x} + \dfrac {\partial f_y} {\partial y} + \dfrac {\partial f_x} {\partial z}\) Definition of Dot Product


for a vector-valued function $\mathbf f = \left({f_x \left({\mathbf x}\right), f_y \left({\mathbf x}\right), f_z \left({\mathbf x}\right)}\right)$.


Also see

  • Results about divergence can be found here.


Sources