DIVERGENCE OF A VECTOR FIELD

The divergence of a vector at a given point in a vector field is a scalar and is defined as the amount  of flux diverging from a  unit  volume element per  second around that point.

The divergence of a vector at a point may be positive if field lines are diverging or coming out from a small volume surrounding the point.

On the other hand, if field lines are converging into a small volume surrounding the point, the divergence of a vector is negative. If the rate at  which field lines are entering  into a small volume  surrounding the  point is  equal to the rate at which these are leaving  that small volume, then the divergence of a vector is zero.

that is, div A = 0.

Analytically

If vector A is the function of x, y and z, then

A = Axi + Ayj + Azk

The operator Λ in cartesian  coordinates is expressed as

\nabla = id/dx + jd/dy + kd/dz

The dot product  of  operator \nabla. A is written as

So divergence of a vector  is a scalar.

\nabla.A = div A = dAx/dx + dAy/dy + dAz/dz

Solenoidal Vector:

Any vector A whose  divergence is zero  is called  solenoidal vector  that is

\nabla.A = div A = 0

CURL OF A VECTOR FIELD

Physical Meaning:

The curl of a vector  at any point is a vector. Curl is a measure of how  much the  vector curls around the point in question.

Analytically:

The curl of a vector A is defined as the vector product or cross product of the  (del) operator and A. Therefore,

Curl of a vector is a vector.

Example. When a rigid body is rotating about a fixed axis, then the curl of the linear velocity of a point on the body  represents twice its angular velocity.

Rotational vector field: Any vector field whose curl is not zero, is called rotational vector field.

Irrotational vector field: Any vector field whose curl is zero, is called irrotational vector field.