divergence

A scalar measuring whether a vector field is, on balance, spreading material away from a point or pulling it in.

For $\mathbf{F}=(P,Q,R)$ , $\operatorname{div}\mathbf{F} = P_x+Q_y+R_z$ . Read as a limit, shrink a closed surface around a point down to that point, and $\operatorname{div}\mathbf{F}$ is the limiting ratio of outward flux to enclosed volume — the flux integral, made local. Divergence is defined pointwise, so a field can be a source in one region and a sink in another; always compute the full sum, never judge from one component alone.

Examples

For $\mathbf{F}(x,y,z) = (x^2,\, -y^2,\, 2z)$ : $\operatorname{div}\mathbf{F} = 2x-2y+2$ , so at $(1,-1,3)$ , $\operatorname{div}\mathbf{F}=6$ , a strong local source.
An incompressible fluid's velocity field satisfies $\operatorname{div}\mathbf{v}=0$ everywhere, with no sources or sinks.