|
(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region ''U''. These are the beginnings and main motivations of [[de Rham cohomology]].
==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 '''R'''<sup>3</sup>.
Define the current two form
:<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>\rho = 1 dx\wedge dy\wedge dz</math> moving with local velocity '''F'''. Its [[exterior derivative]] <math>d j</math> is then given by
:<math>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</math>
Thus, the divergence of the vector field '''F''' can be expressed as:
:<math> \nabla \cdot \mathbf{F} = \star {\mathbf d} \star \mathbf{F}^\flat </math>
Here the superscript <math> \flat </math> is one of the two [[musical isomorphism]]s, and <math> \star </math> 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.
==იხილეთ აგრეთვე==
| 