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.
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
= id/dx + jd/dy + kd/dz
The dot product of operator . A is written as
So divergence of a vector is a scalar.
.A = div A = dAx/dx + dAy/dy + dAz/dz
Any vector A whose divergence is zero is called solenoidal vector that is
.A = div A = 0
CURL OF A VECTOR FIELD
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.
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.