DIVERGENCE OPERATOR

I. Definitions

I.1. Operator part: ∂

	We define the vector partial: ∂ as follows:
	∂ =  ( ∂/∂x, ∂/∂y, ∂/∂z) = (∂x, ∂y, ∂z)
	

I.2. Divergence

	We define the  Divergence  of a vector R = (Rx, Ry, Rz) as
	follows:
	Div (R) = ∇(R) = ∂Rx/∂x + ∂Ry/∂y + ∂Rz/∂z = ∂ . R
	(dot product: Partial . Vector).
	The result is a scalar.

	

II. Applications

II.1. Relationship between divergence and flux

We want to demonstrate that the divergence of a vector (for instance the electric field E) is equal to the rate of the related flux and the related volume.

We can write:
div(E) = dE_x/dx + dE_y/dy + dE_y/dz
dV . div(E) = dx.dy.dz.div(E) =
dE_xdy.dz + dE_ydx.dz + dE_ydx.dy
= flux of E_x through dy.dz element + flux of E_y through dx.dz element
+ flux of E_z through dx.dy element =
flux of dE through the wrapping surface
(as in a box).

	dV.div(E) = dV. ∇. E = dS.dE  
	Hence:
	 div(E)  = ∇. E = flux/ volume