# divergence and curl of vector field

**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** = A_{x}i + A_{y}j + A_{z}k

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 = dA_{x}/dx + dA_{y}/dy + dA_{z}/dz

**Solenoidal Vector:**

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

.**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.

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