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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 $ P



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  automatically generated 8/ 1/2011