Curl and Divergence Theorems of a Vector.ppt

Curl and Divergence of a Vector

1. 3.3-1
Edward C. Jordan Memorial Offering of the First Edward C. Jordan Memorial Offering of the First
Course under the Indo-US Inter-University Course under the Indo-US Inter-University
Collaborative Initiative in Higher Education and Collaborative Initiative in Higher Education and
Research: Electromagnetics for Electrical and Research: Electromagnetics for Electrical and
Computer EngineeringComputer Engineering
byby
Nannapaneni Narayana RaoNannapaneni Narayana Rao
Edward C. Jordan Professor of Electrical and Computer EngineeringEdward C. Jordan Professor of Electrical and Computer Engineering
University of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign
Urbana, Illinois, USAUrbana, Illinois, USA
Amrita Viswa Vidya Peetham, CoimbatoreAmrita Viswa Vidya Peetham, Coimbatore
July 10 – August 11, 2006July 10 – August 11, 2006

2. 3.3
Curl and Divergence

3. 3.3-3
Maxwell’s Equations in Differential Form
Curl
Divergence
=
=
t
t




B
×E
D
×H J
 
 
A A A
x y z
x y z
x y z
a a a
×Α 
∂ ∂ ∂
∂ ∂ ∂
D
B
AA A
=
yx z
x y z
 
  
A  

4. 3.3-4
Basic definition of curl
×A is the maximum value of circulation of A per
unit area in the limit that the area shrinks to the point.
Direction of is the direction of the normal
vector to the area in the limit that the area shrinks
to the point, and in the right-hand sense.
×A
max
Lim
S 0 S
C
n
d
 
A l
×A= a
 
 
 
 


5. 3.3-5
Curl Meter
is a device to probe the field for studying the curl of the
field. It responds to the circulation of the field.

6. 3.3-6

7. 3.3-7
0
0
2
for 0
2
2
2 for
2
z
z
x a
v x
a
x a
v x a
a
a
v
a

 



 
   
 
 
 
negative for 0
2
positive for
2
y
a
x
a
x a
×v

 

   
  

0
0
2
2
0 0
x y z
y
z
y
y
z
v
v a
vx y z x
av
  
  


a a a
a
×v a
a




    



8. 3.3-8
Basic definition of divergence
Divergence meter
is the outward flux of A per unit volume in the limit that
the volume shrinks to the point.
is a device to probe the field for studying the divergence
of the field. It responds to the closed surface integral of
the vector field.
Lim
0v v 
A d S
A=





9. 3.3-9
Example:
At the point (1, 1, 0)
Divergence zero
Divergence positive
Divergence negative
(a)
(b)
(c)

2
1
x
x a
1
y
ya
y
x
y
a
x
y
1
z 1
y
1
z 1
x
y
1
z 1
x

10. 3.3-10
Two Useful Theorems:
Stokes’ theorem
Divergence theorem
A useful identity
 
C S
d dA l= ×A S 
 
×A  
 
S V
d dvA S= A 
 

11. 3.3-11
x y z
x y z
x y z
A A A
  
  
a a a
×Α 
     
0
x y z
x y z
x y z
x y z
x y z
A A A
  
  
  
  
  
  
×A = ×A ×A ×A      
 