Editing Divergence (section)

{{Short description|Vector operator in vector calculus}}
{{About|divergence in vector calculus|divergence of infinite series|Divergent series|divergence in statistics|Divergence (statistics)|other uses}}
{{Calculus|Vector}}
[[File:Divergence (captions).svg|500px|thumb|upright=1.75|alt= A vector field with diverging vectors, a vector field with converging vectors, and a vector field with parallel vectors that neither diverge nor converge|The divergence of different vector fields. The divergence of vectors from point (''x'',''y'') equals the sum of the partial derivative-with-respect-to-''x'' of the ''x''-component and the partial derivative-with-respect-to-''y'' of the ''y''-component at that point:
<math>\nabla\!\cdot(\mathbf{V}(x,y)) = \frac{\partial\, {V_x(x,y)}}{\partial{x}}+\frac{\partial\, {V_y(x,y)}}{\partial{y}}</math>]]

In [[vector calculus]], '''divergence''' is a [[vector operator]] that operates on a [[vector field]], producing a [[scalar field]] giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to area.) More precisely, the divergence at a point is the rate that the flow of the vector field modifies a volume about the point ''in the limit'', as a small volume shrinks down to the point.

As an example, consider air as it is heated or cooled. The [[velocity]] of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.