2.12: Divergence and Curl
The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the vector field is "contracting" or "converging" towards that point. This implies that the vector field is flowing inwards towards the point more than it is flowing outwards. This point is often called the "sink" of the field.
The divergence is zero if the inward flux at a point equals the outward flux. Mathematically, divergence is the dot product of the del operator with the vector field and is expressed as
The curl of a vector field is the circulation of the vector per unit area as this area tends to zero, and is in the direction normal to the area where the circulation is maximum. The curl of a vector field indicates the local rotation or circulation of the vector field calculated at any arbitrary point. A zero curl indicates no rotation, while a non-zero curl indicates rotation of the vector field. Mathematically, curl is the cross product of the del operator with the vector field and is expressed as
Curl is an important concept in many areas of physics, including electromagnetism and fluid dynamics. In electromagnetism, the curl of electric and magnetic fields determines the behavior of electromagnetic waves. Meanwhile, in fluid dynamics, the curl of the velocity field determines the degree to which a fluid "circulates" or "rotates" at a given point.
A curl indicates direction of a non-uniform flow, whereas divergence of the field only shows the scalar distribution of its sources.