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, a…
The divergence of a vector field at a point is the net outward flux per unit volume, as the volume at that point shrinks to zero.
Mathematically, divergence is the dot product of the del operator with the vector field.
Consider a vector field of water flowing through a pipe. If the water flows with constant velocity, it does not diverge.
On passing through a hole, the velocity of water diverges, resulting in a positive divergence.
On connecting the pipe to a multi-holed connector, the water velocity decreases, leading to negative divergence.
The curl of a vector field is the circulation of the vector per unit area, as this area shrinks to zero. It is directed normal to the area where the circulation is maximum. Mathematically, it is the cross product of del operator with the vector field.
Consider a non-uniform velocity vector of a river. A stick tossed into the river floats smoothly where the velocity vector has zero curl, and rotates where the velocity vector has non-zero curl.
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Q1: What does divergence measure in a vector field?
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.
Q2: How is divergence calculated mathematically?
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.
Q3: What is curl and how does it differ from divergence?
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.
Q4: How is curl mathematically expressed?
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.
Q5: What does positive divergence indicate in a flowing fluid?
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.
Q6: What does negative divergence indicate in a flowing fluid?
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.
Q7: How does curl apply to fluid dynamics?
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.