Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.In physical terms, the divergence of a three-dimensional vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its 'outgoingness' – the extent to which there is more of some quantity exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point then there is compression or expansion at that point. (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flux and so on.)In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k {displaystyle mathbf {F} =F_{x}mathbf {i} +F_{y}mathbf {j} +F_{z}mathbf {k} }   is defined as the scalar-valued function:It can be shown that any stationary flux v(r) that is at least twice continuously differentiable in R3 and vanishes sufficiently fast for |r| → ∞ can be decomposed into an irrotational part E(r) and a source-free part B(r). Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl):The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.,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 asThe divergence of a vector field can be defined in any number of dimensions. If