In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above. Gradient in math Directional derivative in math Divergence in math Curl in math Gradient , Directional Derivative , Divergence , Curl in mathematics Gradient , Directional Derivative , Divergence , Curl in math Gradient , Directional Derivative , Divergence , Curl

2. • Operator is called vector differential operator
defined as

ˆˆ ˆi j k
x y z
   
    
   

3. 
( , , )x y z• If is a scalar function of three variables and
is differentiable, the gradient of is defined as
Where ,
is a scalar function
is a vector function
ˆˆ ˆgrad i j k
x y z
  
 
  
    
  



4. If ,determine at point P=(1,3,2).
solution
2 3 2 2
x yz xy z   grad
     
2 3 2 2
3 2 2 2 3 2 2 2 2
3 2 2 2 3 2 2 2 2
2 , 2 , 3 2
,
ˆˆ ˆ
ˆˆ ˆ2 2 3 2
(1,3,2)
ˆˆ ˆ84 32 72
x yz xy z
xyz y z x z xyz x yz xy z
x y z
Therefore
i j k
x y z
xyz y z i x z xyz j x yz xy z k
at
P
i j k

  
  


 
  
     
  
  
   
  
     

   

5. If A and B are two scalars ,then
1)
2)
( )A B A B     
( ) ( ) ( )AB A B B A    

6. Directional derivative of in the direction of a is
ˆ.
d
a grad
ds


where,
Which is a unit vector in the direction of .dr
ˆ
dr
a
dr


7. Compute the directional derivative of at the
point (1,2,-1) in the direction of the vector A=2i+3j-4k.
Solution
2 2 2
2x z xy yz   
Directional derivative of in the direction of a
Where,
And ,

ˆ.
d
a grad
ds


ˆˆ ˆgrad i j k
x y z
  
 
  
    
  
ˆ
A
a
A

2 2 2
2x z xy yz    Hence ,
     2 2 2 ˆˆ ˆ2 2 4 2
(1,2, 1),
ˆˆ ˆ6 9 3
xz y i xy z j x yz k
At
i j k


      

   

8. Also given ,thenˆˆ ˆ2 3 4A i j k  
 
 
22 2
2 3 4 29
,
1 ˆˆ ˆˆ 2 3 4
29
,
ˆ.
51
29
A
Therefore
A
a i j k
A
then
d
a grad
ds


    
   



9. If ,the divergence of A is
defined as
ˆˆ ˆ
x y z
A i j ka a a  
 
.
ˆ ˆˆ ˆ ˆ ˆ.
.
x y z
yx z
divA A
i j k i j k
x y z
x y z
a a a
aa a
 
   
     
   
 
  
  

10. If , determine at point (1,2,3) .
solution
2 2 ˆˆ ˆA x yi xyzj yz k   divA
 
.
ˆ ˆˆ ˆ ˆ ˆ.
.
x y z
yx z
divA A
i j k i j k
x y z
x y z
a a a
aa a
 
   
     
   
 
  
  
2 2divA xy xz yz  
(1, 2,3)
2(1)(2) (1)(3) 2(2)(3)
13
at
divA
divA
  


11. If ,the curl of A is defined byˆˆ ˆ
x y z
A i j ka a a  
 ˆ ˆˆ ˆ ˆ ˆ
ˆˆ ˆ
x y z
x y z
curlA A
i j k i j k
x y z
i j k
curlA
x y z
a a a
a a a
 
   
      
   
  
 
  

12. If ,determine curl A
at (1,3,-2).
solution
4 2 2 2 2 2 ˆˆ ˆ( ) ( )A y x z i x y j x yzk    
4 2 2 2 2 2
2 2 3
ˆˆ ˆ
ˆˆ( 2 2 ) (2 4 )
(1,3, 2),
ˆˆ ˆ2 8 106
i j k
curlA A
x y z
y x z x y x yz
x zi xyz x z j x y k
At
curlA i j k
  
  
  
  
      

  