Editing Divergence (section)

==In arbitrary finite dimensions==

The divergence of a vector field can be defined in any finite number <math>n</math> of dimensions. If 
:<math>\mathbf{F} = (F_1 , F_2 , \ldots F_n) ,</math>

in a Euclidean coordinate system with coordinates {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>}}, define

:<math>\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \cdots + \frac{\partial F_n}{\partial x_n}.</math>

In the 1D case, {{math|'''F'''}} reduces to a regular function, and the divergence reduces to the derivative.

For any {{math|''n''}}, the divergence is a linear operator, and it satisfies the "product rule"

:<math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F})</math>

for any scalar-valued function {{mvar|φ}}.