Mathematics-Online lexicon: Divergence

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	Mathematics-Online lexicon:
	Divergence

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The divergence of a differentiable vector field

$\displaystyle \Phi (x_1, \ldots, x_n) = \left( \begin{array}{c} f_1(x_1, \ldots , x_n) \\ : \\ f_n(x_1, \ldots , x_n) \end{array} \right)$

is defined as

$\displaystyle \operatorname{div} \Phi = \partial_{x_1} f_1 + \ldots + \partial_{x_n} f_n \ .$

In particular for a vector field $\Phi (x,y,z) = \left( \begin{array}{c} p(x,y,z) \\ q(x,y,z) \\ r(x,y,z) \end{array} \right)$ in $\mathbb{R}^3$

the divergence is

$\displaystyle \operatorname{div} \Phi = p_x + q_y + r_z \ .$

Note that the divergence is a scalar function. It is invariant under orthogonal coordinate transformations.

Physical interpretation: The divergence of $\Phi$ is the rate of outward flow per unit volume per unit time at

()

Example:

automatically generated 8/ 1/2011