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

.

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.

Read more about this topic:  Divergence

Famous quotes containing the words relation, exterior and/or derivative:

    [Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.
    —D.H. (David Herbert)

    It’s not a pretty face, I grant you. But underneath its flabby exterior is an enormous lack of character.
    Alan Jay Lerner (1918–1986)

    When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.
    Wyndham Lewis (1882–1957)