Lecture (4) Divergence, Curl Vector Field and Laplacian

Divergence of a Vector Field
Basic Concept of the Divergence:

Fig (1 – 18 ) Divergence of a Vector Field z
(1-59)



Divergence Operator in Cylindrical and Spherical Coordinate Systems
1 – Cylindrical Coordinate System

2 – Spherical Coordinate System

Divergence Theorem
(1-60)
1 – Solenoidal , if ?.E=0
2 – Distributive , because of ? .( E_1+ E_2 )= ? .E_1+ ? .E_2
3 – For a constant E , the entering and leaving fluxes are the same and the divergence is zero , the field is thus divergenceless.
1.11 Curl of a Vector Field
Basic Concept of the Curl
Circulation = ???B .dI? , Circulation of uniform field is zero , for instance, in the case of ( a )
,see fig. ( 1 – 19 a )

Fig (1 – 19 a) Uniform field

Fig.(1 -19 b ) Azimuthal field
For case ( b ) , we have
(1- 61 )
So we have

?_C??B .dI= ?_0 I?

(1-62)




Vector Identities Involving the Curl
1 – Distributive property of two vector fields

2 – Divergence of a curl of a vector field

3- Curl of a gradient of a scalar field

Curl operator in Cylindrical and Spherical Coordinate Systems
1 – Cylindrical Coordinate System

2 – Spherical Coordinate System


Stokes’s Theorem
The Stokes’s Theorem converts the integral of the curl of a vector over an open surface S in to a line integral of the vector along the contour C bounding the surface S.
(1- 63 )
If ? x B=0 , then the field is said to be conservative or irrotational.

1.12 Laplacian Operator
1 – Laplacian Operator in Cartesian Coordinates


2 – Laplacian Operator in Cylindrical Coordinates

3 – Laplacian Operator in Spherical Coordinates