Divergence - Relation With The Exterior Derivative

Relation With 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 R3. Define the current two form

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It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density moving with local velocity F. Its exterior derivative is then given by

d j = \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

Thus, the divergence of the vector field F can be expressed as:

Here the superscript is one of the two musical isomorphisms, and is the Hodge dual. Note however that working with the current two form itself 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.

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