Divergence and Curl

Divergence measures the net outward flux per unit volume at a point as the volume shrinks to zero. It indicates how much a vector field spreads out or diverges from a given point. Positive divergence represents outgoing flux from a source, while negative divergence represents inward flow toward a sink. Zero divergence means inward and outward flux are equal.

Divergence is mathematically calculated as the scalar product dot product of the del operator with the vector field. This operation produces a scalar value representing the field's spreading tendency at any point. The del operator acts on each component of the vector field to yield the divergence result.

Curl measures the local rotation or circulation of a vector field per unit area, directed normal to the area of maximum circulation. Unlike divergence, which shows scalar distribution of sources, curl indicates the direction of non-uniform flow and rotational behavior. Zero curl means no rotation, while non-zero curl indicates the field is rotating at that point.

Curl is mathematically expressed as the vector product cross product of the del operator with the vector field. This operation produces a vector perpendicular to the plane of circulation. The resulting vector points in the direction of maximum circulation and has magnitude proportional to the rotation strength.

Positive divergence indicates that fluid is flowing outward from a point, which is called a source. For example, water flowing through a hole in a pipe shows positive divergence because the velocity increases as it exits. The outgoing flux exceeds the incoming flux at that location.

Negative divergence indicates that fluid is flowing inward toward a point, called a sink. When a pipe connects to a multi-holed connector, water velocity decreases as it converges, resulting in negative divergence. More fluid flows inward than outward at that location.

In fluid dynamics, curl of the velocity field determines the degree to which a fluid circulates or rotates at a given point. A stick floating in a river rotates where the velocity vector has non-zero curl, indicating local rotation. Where curl is zero, the stick floats smoothly without rotating, showing no circulation.