This document provides an overview of Dirac delta functions and their properties. It defines the Dirac delta function as equal to 0 for all values of x except when x is 0, where it is equal to infinity. The key properties listed are that the Dirac delta function can be used to sample or shift other functions, replicate functions, and behave like the derivative of the Heaviside step function. Integrating a function multiplied by the Dirac delta returns the value of the function at 0.

1. Series: EMF Theory
Lecture: # 0.15
Dr R S Rao
Professor, ECE
Del applied to certain functions, product rules, second derivatives
Dirac delta functions, its properties.
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Teaching
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Learning

2. The coordinates of the origin are::
(0,0,0)in Cartesian system
(0,f,0)in in Cylindrical system
(0,θ,f)in in Spherical system
The coordinates of a point right over the z-axis at a height of z are::
(0,0,z)in Cartesian system
(0,f,z)in Cylindrical system
(z,0,f)in Spherical system
Electromagnetic
Fields
VECTOR
CALCULUS-II
2
Certain Aspects

3. Let us consider two points,P1 and P2whose coordinates are,
•(x1,y1,z1)and (x2,y2,z2)in Cartesian system,
•(ρ1, f1,z1)and (ρ2, f2,z2)in Cylindrical system,
•(r1, θ1, f1)and (r2,θ2,f2)in Spherical system
Electromagnetic
Fields
VECTOR
CALCULUS-II
3
     
2 1 2 1 2 1
ˆ ˆ ˆ
D x y z
x x y y z z
     
     
2 2 2
2 1 2 1 2 1
d x x y y z z
      
D
In Cartesian system they can be found to be,
Certain Aspects

4. In other coordinate systems, first the coordinates of the points need
to be converted into Cartesian system, and then, the above formulae
are to be applied.
Electromagnetic
Fields
VECTOR
CALCULUS-II
4
In cylindrical system they can be found to be
     
2 2 1 1 2 2 1 1
ˆ ˆ
cos cos sin sin
 f  f  f  f
     
D x
     
2 1
ˆ ˆ
z z
 f  f  f  f
     
y z
ŷ
   
2
2 2
2 1 1 2 2 1 2 1
2 cos
d z z
    f f
     
In spherical system they can be found to be
 
2 2 2 1 1 1
ˆ
sin cos sin cos
r r
 f  f
 
D x  
2 2 2 1 1 1
sin sin sin sin
r r
 f  f
  y  
2 2 1 1
ˆ
cos cos
r r
 
  z
 
2 2
2 1 1 2 1 2 1 2 2 1
2 cos cos sin sin cos
d r r rr     f f
 
    
 
Certain Aspects

5. 5
2 2 2 1 2
(1). [( ) ( ) ( ) ]
ˆ
(2).
ˆ
Note,when ( , , ) (0,0,0), , and
is position vector of point ( , , ).
R x x y y z z
R
x y z R r r
x y z
  
    
   
=
R = R
= r = r
R = r
Electromagnetic
Fields
Vector
Calculus-IV Del Application
The following vector and a scalar functions are faced very often in em field theory.
The gradient, divergence and curl are,
2
2
2
ˆ
1
3
1
1
ˆ
(1). = & ( )=-
ˆ
(2). ( ) =4 ( )
ˆ
(3). ( ) =0
R R
R
R
R
R

 


R
R
R
R

6. 6
Product rules:
Notice, first two are gradients, next two are divergences and the last two are curls.
The application of del operator, on (i) the product of two scalar functions, (ii) two
vector functions and (iii) a scalar function with a vector function, is often faced in
electromagnetics.
(1). ( ) ( ) ( )
fg f g g f
    
(2). ( ) ( ) ( ) ( ) ( )
          
A B A B B A A B B A
(3). ( ) ( ) ( )
f f f
    
A A A
(4). ( ) ( ) ( )
     
A B B A A B
(5). ( ) ( ) ( )
f f f
     
A A A
(6). ( ) ( ) ( ) ( ) ( )
          
A B B A A B A B B A
Electromagnetic
Fields
Vector
Calculus-IV
Certain Identities

7. 7
Second derivatives:
• First one implies a divergence-less vector can always be expressed as curl of an
arbitrary vector function.
• Last one implies a curl-less vector can always be expressed as gradient of an
arbitrary scalar function.
(1). ( ) 0
  
A
2
(2). ( ) ( )
      
A A A
2
(3). ( )
f f
   
(4). ( ) 0
f
  
Electromagnetic
Fields
Vector
Calculus-IV
Certain Identities

8. Impulse is usually indicated by an upwards arrow with its strength by its side.
Dirac delta or unit impulse is not an ordinary function; it is defined as
Electromagnetic
Fields
VECTOR
CALCULUS-II DIRAC DELTA FUNCTION
8
( ) 0 at 0 and ( ) 1
x x x dx
 


  

It can be viewed as an ordinary pulse function with its width tending to zero while
maintaining the area fixed.

9. Electromagnetic
Fields
VECTOR
CALCULUS-II DIRAC DELTA FUNCTION
9
It's important properties are listed below:
 ( ) ( ) ( )
o o
f x x x dx f x



 
  Sampling/shifting property.
 ( ) ( ) ( ) ( ) ( )
o o o
f x x x dx f x x f x
 


   
  Replication property.
 ( ) ( )
x x
 
 
 1
( ) ( )
a
ax x
 

 ( ) ( )
x x x
 
  
  
( ) ( ) 0 ( )
f x x f x
 

  
( ) ( ) 0
f x x dx f



 



10. ENOUGH
FOR
TODAY
10
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY