Vector operators
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The document defines and provides properties of gradient, divergence, and curl - three important vector operators in multi-variable calculus. Gradient is defined as the maximum rate of change of a scalar function in space. Divergence measures how a vector field spreads out from a point, with zero divergence indicating a solenoidal (non-spreading) field. Curl measures the maximum rotation of a vector field around a point, with zero curl indicating an irrotational (non-rotating) field. Stokes' theorem relates the circulation of a vector field around a closed path to the surface integral of the curl over the enclosed surface.
Vector operators
- 2. The DEL ( ) is the , vector partial differential , an operator In Cartesian coordinates Gradient 1.When the DEL is operated on scalar 2.Gadiet of the scalar is a vector 3.It gives maximum rate of change of the function in space 4.If V is the scalar function then gradient of V ( ): DEL ( ) operator
- 3. If find Gradient of V ( ) at a point P(2,1,1) At point P
- 4. 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
- 5. If , find divergence ( ) of D at a point P(2,3-1). We know that At point ‘P’
- 6. 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
- 7. Curl 1.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
- 8. Curl 1.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
- 9. If find Stokes theorem Stokes 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.