Editing Divergence (section)

==Relation to the exterior derivative==
One can express the divergence as a particular case of the [[exterior derivative]], which takes a [[2-form]] to a 3-form in {{math|'''R'''<sup>3</sup>}}. Define the current two-form as
:<math>j = F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy .</math>
It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density {{math|''ρ'' {{=}} 1 ''dx'' ∧ ''dy'' ∧ ''dz''}} moving with local velocity {{math|'''F'''}}. Its exterior derivative {{math|''dj''}} is then given by
:<math>dj = \left(\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx \wedge dy \wedge dz = (\nabla \cdot {\mathbf F}) \rho </math>
where <math>\wedge</math> is the [[wedge product]].

Thus, the divergence of the vector field {{math|'''F'''}} can be expressed as:
:<math>\nabla \cdot {\mathbf F} = {\star} d{\star} \big({\mathbf F}^\flat \big) .</math>
Here the superscript {{music|flat}} is one of the two [[musical isomorphism]]s, and {{math|⋆}} is the [[Hodge star operator]]. When the divergence is written in this way, the operator <math>{\star} d{\star}</math> is referred to as the [[codifferential]]. Working with the current two-form and the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes with a change of (curvilinear) coordinate system.