Divergence of a Vector Field

The divergence of a vector field at any point is defined as the amount of flux per unit volume diverging from that point.

Let v (x, y, z) represents a vector field at a point (x, y, z) in a certain region of space. The divergence of v is defined as the dot product of â–¼ and v i.e., â–¼.v. It is a scalar quantity.
            â–¼.v = (i ∂/∂_x + j ∂/∂_y + k ∂/∂_z) (iv_x + jv_y + kv_z)
                = ∂v_x /∂_x + ∂v_y ∂_y + ∂v_z /∂_z

Physical Significance of Divergence. In problems of hydrostatics, let v (velocity of a fluid) be the vector field.

(i)    If divergence of v is positive, it indicates a net outward flow of fluid from the point. This means either the fluid is expanding (resulting in the corresponding decrease in density) or the given point is a source points.
 
(ii)    If divergence of v is negative, it indicates a net flow towards the point. This means either the fluid is contracting (resulting in corresponding increase in density) or the given point is a ‘sink point’.

(iii)    If divergence of v is zero, there is no net flow of fluid across any surface. This means the fluid is neither expanding nor contracting, In other words, the fluid is incompressible.

In the case of non-material fluxes (such as electric or magnetic flux), the existence of divergence means the presence of source or sink of flux at the point. In an electric field, for example, positive charges constitute source of flux while negative charges constitute sink. Therefore, if the divergence at a point in an electric field is positive, it means there are positive charges at that point. If the divergence is negative, it means the presence of negative charges at that point.