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The Curl of a Vector Field

The vector product or cross product of the differential operator ‘\1 with a vector is known as the curl of that vector,

Thus,

Curl A =∆ x A = |I ∂ /∂x

 

Where ,A =I A x + j A y +k A

Hence, curl of a vector point function A is the vector product of the del operator with A.

The curl of a vector A may be expressed as

 

If curl of a vector field F about a point may also be defined as the circulation per unit surface as the surface shrinks to zero. In terms of line integral it is defined as

where n is the unit vector normal to area ∆S.

 

This equation gives the component of the curl which is in the direction of unit normal to ∆S. The net circulation or rotation of the field will be a vector resultant of the circulation in the three orthogonal planes.

 

Physical Significance of Curl the Curl in a Cartesian Co-ordinates

Now considering an area of simple shape, ABCD in the x-y plane. Let the sides

AB = CD =∆x and BC =AD =∆y

 

Let  Fx and F y be the components of F along AB and AD respectively. Then

The line integral over a closed cycle ABCD, therefore, is given by Combining the components,