Editing Divergence (section)

=== Cylindrical coordinates ===
For a vector expressed in '''local''' unit [[Cylindrical coordinate system|cylindrical coordinates]] as
<math display="block">\mathbf{F} = \mathbf{e}_r F_r + \mathbf{e}_\theta F_\theta + \mathbf{e}_z F_z,</math>
where {{math|'''e'''<sub>''a''</sub>}} is the unit vector in direction {{math|''a''}}, the divergence is{{refn|[http://mathworld.wolfram.com/CylindricalCoordinates.html Cylindrical coordinates] at Wolfram Mathworld}}
<math display="block">\operatorname{div} \mathbf F = \nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} \left(r F_r\right) + \frac1r \frac{\partial F_\theta}{\partial\theta} + \frac{\partial F_z}{\partial z}.
</math>

The use of local coordinates is vital for the validity of the expression. If we consider {{math|'''x'''}} the position vector and the functions {{math|''r''('''x''')}}, {{math|''θ''('''x''')}}, and {{math|''z''('''x''')}}, which assign the corresponding '''global''' cylindrical coordinate to a vector, in general {{nowrap|<math>r(\mathbf{F}(\mathbf{x})) \neq F_r(\mathbf{x})</math>,}} {{nowrap|<math>\theta(\mathbf{F}(\mathbf{x})) \neq F_{\theta}(\mathbf{x})</math>,}} and {{nowrap|<math>z(\mathbf{F}(\mathbf{x})) \neq F_z(\mathbf{x})</math>.}} In particular, if we consider the identity function {{math|1='''F'''('''x''') = '''x'''}}, we find that:

<math display="block">\theta(\mathbf{F}(\mathbf{x})) = \theta \neq F_{\theta}(\mathbf{x}) = 0.</math>