vector operators

The DEL ( ) is the , vector partial differential , an operator In Cartesian coordinates Gradient1.When the DEL is operated on scalar2.Gadiet of the scalar is a vector3.It gives maximum rate of change of the function in space4.If V is the scalar function then gradient of V ( ):

DEL ( ) operator

If find Gradient of V ( ) at a point P(2,1,1)

At point P

Divergence 1.Divergence of a vector function ( ) is equal to the dot product of and

( ) 2 .( The physical interpretation of , divergence of electric flux density :) The divergence of the vector field ( )is the outflow of flux from a small closed

surface per volume as the volume shrink to zero.

3.If the divergence of a vector field is zero, such a field is called as solenoidal field

4. If

If , find divergence ( ) of D at a point P(2,3-1).

We know that

At point ‘P’

Divergence Theorem The surface integral of, the normal component of, any vector field over a closed surface is equal to the volume integral of the divergence of this vector field throughout the volume enclosed by the closed surface

Curl1.Curl of a vector function( )is equal to cross product of and 2.Curl of a vector field is a vector . 3. Curl of a vector field gives the maximum rotation of vector field per

unit area

3. If

5.If the curl of the vector field is zero then it is called as irrotational field

Curl1.Curl of a vector function( ) is equal to cross product of and ( ) 2.( The physical interpretation of curl of vector field :) Curl of a vector field gives the maximum rotation of vector field per unit

area as the area tends to zero.

3.If the curl of the vector field is zero then it is called as irrotational field

3. If

If find

Stokes theoremStokes theorem states that the circulation of a vector field ,around a closed path, is equal to the surface integral of the curl of that vector field, over the open surface bounded by the closed path.