In vector calculus, divergence is a vector operator that operates on a vector field, producing a

scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantit ...

giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward

flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...

of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

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.

Physical interpretation of divergence

In physical terms, the divergence of a 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 are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence. The divergence of a vector field is often illustrated using the simple example of the

velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...

of a fluid, a liquid or gas. A moving gas has a

velocity Velocity is the dire