Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
UNIT # 04
GRADIANT DIVERGENCE AND CURL Introduction: In this chapter, we will discuss about partial derivatives, differential operators Like Gradient of a scalar ,Directional derivative , curl and divergence of a vector . Partial Derivative: Let ⃗ be a vector function of independent scalar variable ⃗=
(
) ̂
(
)̂
⃗ ⃗
⃗
)̂
(
Then 1st 0rder partial derivatives w .r . t
as
are define as
=
( ) ̂
( )̂
( )̂
(
behave as a constant)
=
( ) ̂
( )̂
( )̂
(
behave as a constant)
=
( ) ̂
( )̂
( )̂
Higher order partial derivatives of ⃗ w .r . t
(
behave as a constant)
are define in a similar way.
The vector Differential Operator Del ( ⃗⃗⃗ ) : A vector ⃗⃗⃗ = ̂
̂ is called Differential Operator Del ( ⃗⃗⃗ ) . ̂
Gradient of a scalar : Let
(
) is a scalar function in a space. Then Gradient of a scalar is define as ; ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗
(
̂
̂) ̂
=
̂
̂
̂
Properties of Gradient : If ( ) ⃗⃗⃗ (
and
are scalar function and c is constant then
) = c ⃗⃗⃗
Proof: We know that ⃗⃗⃗ = ̂
̂
̂
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Page 1
Vector Analysis: Gradient, Divergence and Curl Then ⃗⃗⃗ (
)=(
( ) ⃗⃗⃗ (
̂
̂
) = ⃗⃗⃗
⃗(
)=(
̂
̂ ̂
̂
̂
̂
̂ ) = c ⃗⃗⃗ ̂
(
(
̂
) ̂
(
) ̂
̂ ) = ⃗⃗⃗ ̂
) ̂
(
⃗⃗⃗
] ̂ (
⃗⃗⃗⃗
̂ ̂
̂ )( ̂
=[
( ) ⃗⃗⃗ ( ) =
(
⃗⃗⃗
)=(
=
̂ ̂
)=
̂) ̂
Proof: We know that ⃗⃗⃗ = Then ⃗⃗⃗ (
̂
̂ )( ̂
⃗⃗⃗
)=
̂
̂
=( ( ) ⃗⃗⃗ (
)=c
⃗⃗⃗
Proof: We know that ⃗⃗⃗ = Then
̂ )(
B.Sc & BS Mathematics
̂
)=
( ] ̂
[ ̂)
̂
) ̂
(
(
) ̂
(
] ̂
[
̂
) ̂
̂)= ̂
⃗⃗⃗
⃗⃗⃗
⃗⃗⃗
Proof: Let ⃗⃗⃗ ( )
⃗⃗⃗ (
⃗⃗⃗ ( )
)
⃗⃗⃗
=
) ⃗⃗⃗
(
⃗⃗⃗
=
⃗⃗⃗
⃗⃗⃗⃗
=
⃗⃗⃗⃗
⃗⃗⃗
Laplacian Operator: If ⃗⃗⃗ {
̂ ⃗⃗⃗ ⃗⃗⃗
̂ ̂ (
̂
Then ̂
= ̂) ( ̂
is called Laplacian Operator. ̂
̂)
}
Laplacian Equation: If f (
) is function then Laplacian Equation is written as
=0
0r
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=0.
Page 2
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
Theorem: Prove that the gradient is a vector perpendicular to the level surface. Proof: Let ⃗ =
̂
̂ ̂
⃗ =d ̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
We have to prove
(
)
Then
d
By using calculus
d
̂
=0 z= 0
̂) ( ̂
̂
̂) ̂ ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ This show that ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
).
⃗
Now as
(
)
be a position vector of any point P on the given surface. Then ̂ is a tangent vector to surface at point P (
̂
(
⃗ ⃗
⃗
Hence , Show that the gradient is a vector perpendicular to level surface at point P( Theorem: Prove that the gradient of a scalar function
(
)
)
is a directional derivative of
perpendicular to the level surface at point P. Proof: Let P & Q be the two neighboring points in a region of space. (
Consider the level surfaces
)
(
&
)
through P & Q respectively. Let the
normal to the level surface through P intersect the level surface through Q at point P. Let ̂ & vectors along ⃗⃗⃗⃗⃗⃗
& ⃗⃗⃗⃗⃗⃗ . = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
We have to prove Let
̂ unit
⃗⃗⃗⃗⃗⃗ Since
& ⃗⃗⃗⃗⃗⃗ =
Applying limit when P
then
̂
⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗
= then
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Page 3
Vector Analysis: Gradient, Divergence and Curl = Here ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
B.Sc & BS Mathematics
|̂||̂|
=
(̂ . ̂ ) = ̂ . ̂
= ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . ̂
=
. It is clear that ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ lies in the directional of normal to the level surface
̂
and measure the rate of change of =(
=
̂
in that direction. ̂) (
̂
̂
̂ ) = ⃗⃗⃗ ̂
̂
Let
= ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ̂
̂
(
Hence proved that the gradient of a scalar function
)
is a directional derivative of
perpendicular to the level surface at point P. . Find ⃗⃗⃗
Example#01: If
|⃗⃗⃗ | at (
).
Solution: Given function We know that ⃗⃗⃗ ̂
⃗⃗⃗
= (
): ⃗⃗⃗
At ( Now
̂= ̂
) ̂
( ( )(
|⃗⃗⃗ |
( )
⃗⃗⃗
(ii)
Solution:
Let ⃗
⃗⃗⃗ ( )
=[ ( ) ] ̂
̂
( )
[
( )
̂
( )⃗⃗
( )[ ̂
̂
̂
)̂= ̂
̂
̂
= √
=√
( ) -----(i) ( )
[
( ) ] ̂
[
(
) ̂
then
( )
̂
( ) ] ̂
= ⃗⃗⃗ ( )
̂
)̂
(
use above result to evaluate the following. (iii)
̂
(
( ) =√
( )⃗⃗
)̂
(
( )̂
) ̂
) ̂
Example#02: Prove that ⃗⃗⃗ ( ) ( )⃗⃗⃗
(
)
)
√(
) ̂
(
] ̂
[
( )
] ̂
[
( )
] ̂
() {
}
̂]
Hence proved.
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Page 4
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
( ) ⃗⃗⃗ Solution: Let
( )=
( )
then
Using given equation.
⃗⃗⃗ ( )
( )⃗⃗
then
( )
⃗⃗⃗ ( )
( )⃗⃗
)⃗⃗
(
⃗⃗⃗
⃗
( ) ⃗⃗⃗ Solution:
( )=
Let
Using given equation. ( )
( )⃗⃗
⃗⃗⃗
⃗
( ) ( )=
Solution: Let
( )
then
(
)
Using given equation ( )⃗⃗
⃗⃗⃗ ( )
)⃗⃗
(
⃗⃗⃗ ( )
⃗ =
( ̂
̂)= ̂
̂
̂
̂
Now ⃗⃗⃗ ⃗⃗⃗ ( )
( )
(
)
̂
̂) ( ̂
=
(
(
)
=
[(
=
[
=
[
{
=
[
{( )
=
[
{
}
=
[
{ }
]= [
̂
( )
̂) ̂
)= [ (
(
) )
(
)
(
(
)] )]
] }
( )
] ()
() }
]=
] {
[
{ ]= [
}
}
] ]= [ ]
( )
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Page 5
Vector Analysis: Gradient, Divergence and Curl
Example#03:If
B.Sc & BS Mathematics
then show that ⃗⃗⃗
is a function of u and u is a function of
Solution: We know that ⃗⃗⃗ ̂
⃗⃗⃗
̂ ̂
By using chain rule of differentiation ; Then
⃗⃗⃗
(
⃗⃗⃗
) ̂
(
) ̂
⃗⃗⃗
=
= ̂
) ̂=
(
(
̂
̂) ̂
Hence proved. such that (i) ⃗⃗⃗
Example#04: Find the scalar function ( ) ⃗⃗⃗
&
= ̂
̂ ̂
(ii) ⃗⃗⃗
=2
⃗
̂ ̂
Solution: We know that ⃗⃗⃗ ̂
Comparing coefficients of ̂ , ̂
̂ ̂
̂
then
̂= ̂ ̂
̂
̂
̂ ∫
=
2∫
=
(
∫
=
(
) --------(i)
(
) --------(ii) )--------(iii)
Adding (i) ,(ii) & (iii) : (
= =[
Hence ( ) ⃗⃗⃗
=2
(
)
(
)
]
⃗⃗
Let ⃗
Solution:
)
̂
We know that ⃗⃗⃗ ̂
̂
̂=2
̂
̂
̂ =(
̂ ̂
̂ ⃗
then ̂
̂ =2
-----(i)
( ̂
then
)
̂)
̂
̂ (
) ̂
(
)
̂
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Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
̂
Comparing coefficients of ̂ , ̂ (
)
∫(
)
=
(
)
∫(
)
=
(
)
∫(
)
=
From (i) ,(ii) & (iii) Example#05: If Solution: Let ⃗
̂
̂ ̂ ⃗⃗⃗
We know that =
⃗⃗⃗
=[
⃗⃗⃗
=[
⃗⃗⃗
=(
)
⃗⃗⃗
=(
)
⃗⃗⃗
=(
)
⃗⃗⃗
=(
(
) ̂ ( ]
) [ ̂
[
̂
̂
[
[ ̂
(
)----(iii)
)
] ̂
) ] ̂
(
[
̂
] ()
̂]
{
}
̂] ̂
̂
[
)
) -----(ii)
⃗.
)
(
[ ]
̂
(
(
)̂
(
] ̂
)
)-----(i)
̂ ̂
)̂
(
(
-----(i)
̂
(
=(
then
=
)
)
.Then show that ⃗⃗⃗
=
⃗⃗⃗
(
=
(
̂] ̂
⃗
)
Hence proved. ⃗⃗
Example #06: If ⃗⃗⃗ Solution: Let ⃗
̂
Then show that ̂ ̂
̂ ̂
̂
̂
̂= ̂ =(
⃗⃗
=
(
(
then
⃗⃗⃗
We know that
( )
̂
̂
⃗
̂ ) ̂
(
( )
)
at
)
-----(i)
̂
then
̂
̂= )
( ̂ ̂
̂
̂)
(
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)
̂ Page 7
Vector Analysis: Gradient, Divergence and Curl ̂
Comparing coefficients of ̂ , ̂ (
)
∫(
)
=
(
)
∫(
)
=
(
)
∫(
)
=
From (i) ,(ii) & (iii):
(
=
)
( )=
Hence ( )=0
At
B.Sc & BS Mathematics
=
(
(
)
(
)
(
(
)
)
)------(i)
(
(
) -----(ii) )-------(iii)
=
------------(a) =0
( )
=0
Hence equation (a) will become ( )= ( )
(
)
Hence proved. ⃗⃗⃗
Example# 07: Show that Solution: Let ⃗
̂
⃗ .
= (n+2) ̂
̂
then
------(i)
Now ⃗⃗⃗
=
⃗⃗⃗
= [(
)
⃗⃗⃗
= [(
)
⃗⃗⃗
=(
)
⃗⃗⃗
=(
)
⃗⃗⃗
=(
)
⃗⃗⃗
=(
)
̂
̂ ̂
] ̂ ] ̂
[
̂
[ ⃗
̂]
] ̂
[(
] ̂
)
[(
] ̂
) ()
{
}
̂] ̂
̂
] ̂
)
[(
̂
̂
[
)
[(
̂
̂]
Hence proved.
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Page 8
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
at (
Example#08: Find a unit vector perpendicular to the surface
).
Solution: Given function We know that ⃗⃗⃗
is perpendicular to the given surface. Therefore
⃗⃗⃗ ̂ (
= ⃗⃗⃗
) ̂ ̂
=
(
At
̂ ̂
(
)̂
)̂
(
̂ ̂
)
⃗⃗⃗
= ( ) ̂
̂= ̂
( )̂
̂ ̂
Now Unit vector of ⃗⃗⃗
=
⃗⃗⃗ |⃗⃗⃗
|
=
̂ ̂ ̂
√
(
=
)
̂ ̂ ̂
√
̂ ̂ ̂
=
√
at (
Example#09:Find the directional derivative of ̂
) in the direction of
̂. ̂
Solution: Given Then ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
= ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=(
At
̂ ) ̂
P(
Let ⃗⃗⃗ =
̂= ̂ (
(
)̂
) ̂
(
)̂
)̂
(
)̂
(
̂
) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=[ ( )
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=[
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=
̂
] ̂ ̂
̂ Then
) ] ̂
[
]̂
]̂
( )( [
̂
( )( ) ] ̂
[
)] ̂
̂
⃗⃗⃗
̂ =|⃗⃗⃗ | =
[
( ) (
̂ √( )
̂ ̂
(
)
( )
=
̂
̂
̂
√
̂
=
̂ ̂
√
=
̂
̂
̂
Thus Directional derivative of
at Point P in the of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂ =(
̂
̂
̂) . (
̂
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̂
̂)
Page 9
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics = =
Example #10: Find the Laplacian equation if f if ( Solution: Given function
(
)=
We know that Laplacian Equation is ( [ [
)
(
)] ]
)=
[
=0
(
=0
)
(
)]
[
0r
]
[
( [
(
)=0 )] = 0
]=0
or This is required equation .
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Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
Exercise# 4.1 Q#01: Find ⃗⃗⃗
.
(i)
(ii) at (
(iii)
( )
)
(
) at (1,1,1)
(i) Solution:
Given function
We know that ⃗⃗⃗ ⃗⃗⃗
̂
̂= ̂
(
) ̂
̂
=
(
)̂
(
)̂
̂ ̂
( ) Solution : Given function We know that ⃗⃗⃗ ̂ (
= ⃗⃗⃗ ( )
̂x ̂
) ̂
= (
) ̂
at (
(
( )̂
)̂
(
)̂
)̂
(
)
Solution : Given function We know that ⃗⃗⃗ ⃗⃗⃗
̂
At ( ⃗⃗⃗
̂
=
⃗⃗⃗
̂= ̂
[
̂
(
(
)̂
(
)̂
̂ ̂
̂
) ̂
̂]
): [( )( ) ̂
( )( ) ̂
( )( ) ̂ ] = ̂
̂
̂
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Page 11
Vector Analysis: Gradient, Divergence and Curl ( )
(
) at (1,1,1) (
Solution : Given function ⃗⃗⃗
We know that ⃗⃗⃗
=
⃗⃗⃗
=
(
⃗⃗⃗
⃗⃗⃗
√
=[
√
[
√
√
=
√
[
√
⃗⃗⃗ Let ⃗ ⃗⃗⃗
)
√
(
)
√
[
√
( ̂
(
̂
√
[
√
√
√
] ̂
]̂
√
√
]̂
[
√
]̂
]̂
√
)[ ̂
√ ̂ Then
̂
)
]̂
√
] ̂
(
] ̂
√
√
√
̂] ̂
] ̂
√
(
√
̂
√
√
√
[
√
)
)
( )[ ̂
̂ ̂
( ̂
(
[
⃗⃗⃗ = [
⃗⃗⃗
√
̂]= ̂
√
) ̂
)̂
(
√
) (
We know that
)̂
)[ ̂
(
) ̂
(
̂] ̂
(
.Where
)
) ̂ (
)[ ̂
) : ⃗⃗⃗
(
(
) ̂
Solution: Given function
=
̂x ̂
) ̂
(
Q#02: Find ⃗⃗⃗
)
̂
(
⃗⃗⃗ At (
B.Sc & BS Mathematics
=√
̂] ̂ &
Thus
)⃗
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Page 12
Vector Analysis: Gradient, Divergence and Curl . Find ⃗⃗⃗
Q#03: If
& |⃗⃗⃗ |
B.Sc & BS Mathematics
(
)
Solution: Given function We know that ⃗⃗⃗ ̂
⃗⃗⃗ At(
̂= ̂
=(
) ⃗⃗⃗
=
⃗⃗⃗
=
)̂
[ ( ) ̂
|⃗⃗⃗ | = √(
(
( )(
(
) ̂
)̂ )] ̂
)
(
Q#04: Find the Laplacian equation if (
)=
(
)
[
√
]̂
[
]̂
]̂
[
z =0
)
(
)]
[
]
= √ z
)=
We know that Laplacian Equation is
[ (
[ ( )( ) ] ̂ =[
[ ( ) ]̂
) =√
(
)̂
(
̂ ̂
Solution: Given function
)̂
)̂
(
(
Now
(
0r
=0 )
(
( )]
[
]
) =0 [ (
)] = 0
[
] =0
=0 =0
0r
such that (i) ⃗⃗⃗
Q#05 : Find the scalar function ()
⃗⃗⃗
= ̂
=0
= ̂
̂
This is required equation.
(ii) ⃗⃗⃗
=
̂
̂
̂
̂
̂
̂
Solution: We know that ⃗⃗⃗ Comparing coefficients of ̂ , ̂
Adding (i) ,(ii) & (iii) :
̂
̂
̂
then
̂= ̂ ̂
̂
̂ ∫
=
(
) ---------(i)
∫
=
(
) --------(ii)
∫
= =
( (
)----------------(iii) )
(
)
(
)
=[
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
]
Page 13
Vector Analysis: Gradient, Divergence and Curl ⃗⃗⃗
( )
̂
=
B.Sc & BS Mathematics
̂ ̂
Solution: We know that ⃗⃗⃗ ̂
̂ ̂
3∫
= ∫
⃗⃗
⃗⃗
(i)
̂
̂
(v) ⃗⃗
̂
̂
̂
(
)
(
)
where
(iii) ⃗⃗ ) ̂
̂
̂
(
)̂
such that ⃗⃗ = ⃗⃗⃗
)̂
(
⃗⃗⃗
The
̂
̂
̂
̂ ̂
̂= ̂
Comparing coefficients of ̂ , ̂
̂
̂
̂
̂ ) --------(i)
∫
=
2∫
=
(
) --------(ii)
∫
=
(
)--------(iii)
Adding (i) ,(ii) & (iii) : Hence
̂
(
̂ ̂
We know that ⃗⃗⃗ Then
)
̂ ̂
Given ⃗⃗
̂
⃗⃗
)----------------(iii)
]
such that ⃗⃗ = ⃗⃗⃗
(ii)
( )
at
(
=[
Q#06 : Find the scalar function
(iv) ⃗⃗
(
=
Hence
) --------------(ii)
(
=
Adding (i) ,(ii) & (iii) :
̂
̂ ̂
) ----------------(i)
( =
∫
̂
̂= ̂ ̂
̂
Comparing coefficients of ̂ , ̂
(i) ⃗⃗
̂
then
(
(
= =[
)
(
)
(
)
]
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 14
Vector Analysis: Gradient, Divergence and Curl ̂
⃗⃗⃗
(ii)
̂
̂
Given ⃗⃗
̂
such that ⃗⃗ = ⃗⃗⃗ ̂
⃗⃗⃗
Then
̂
We know that ⃗⃗⃗ ̂
Then
B.Sc & BS Mathematics
̂
=
̂
̂ ̂
̂= ̂
̂
̂
̂ ̂
Comparing coefficients of ̂ , ̂ ∫
(
=
∫
=
) (
) --------(i)
( )
(
) -------(ii)
From (i) & (ii) (
= ⃗⃗⃗
(iii)
̂
̂
Given ⃗⃗ Then
̂
⃗⃗⃗
̂ ̂
̂
̂
such that ⃗⃗ = ⃗⃗⃗ ̂
̂
We know that ⃗⃗⃗ Then
)
̂ ̂= ̂
̂
̂
Comparing coefficients of ̂ , ̂ ∫ ∫
=
(
) --------(i)
=
(
) -------(ii)
From (i) & (ii) =
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 15
Vector Analysis: Gradient, Divergence and Curl ⃗⃗
⃗⃗⃗
(iv)
̂ ̂
̂ ⃗⃗
Given ⃗⃗
( )
at
Solution: Let ⃗
̂ ̂
(Example #06) ̂
̂
⃗⃗
(
then
⃗
=
)
⃗⃗ = ⃗⃗⃗
such that ̂=
B.Sc & BS Mathematics
̂ =(
) ̂
̂= ̂
(
⃗⃗
⃗⃗⃗
Then
̂
-----(i)
)
( ̂ ( ̂
(
)
∫(
)
=
(
)
∫(
)
=
(
)
∫(
)
=
=
)
̂
̂
Comparing coefficients of ̂ , ̂
From (i) ,(ii) & (iii):
̂) ̂
(
=
(
)
(
)
(
)
)-------(i)
(
( (
) -----(ii) )-------(iii)
)
(
)
= Hence At
( )= ( )=0
---------- --(a)
( )
=0
=0
Hence equation (a) will become ( )=
( )
(
)
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 16
Vector Analysis: Gradient, Divergence and Curl ⃗⃗⃗
(v)
) ̂
(
Given ⃗⃗
) ̂
⃗⃗⃗
We know that
⃗⃗⃗ ̂
̂
̂ =(
̂
Then
(
(
) ̂
)̂
)̂
(
̂ (
)̂
)
=
)̂
(
̂ ∫(
)
(
) ̂ such that ⃗⃗ = ⃗⃗⃗
(
)̂
) ̂
)
(
)̂
(
(
̂
Comparing coefficients of ̂ , ̂ (
)̂
(
(
Then
B.Sc & BS Mathematics
∫(
)
∫(
) ----------(i)
(
)
= 3y+
)
=
) ---(ii)
( (
)--------(iii)
Adding (i) ,(ii) & (iii) (
= =[
̂
̂= ̂
)
⃗⃗⃗
̂ √( )
) in the direction of ⃗⃗⃗⃗⃗⃗ where Q
at (
) ̂
=(
) ̂
(
)̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
= [ ( )] ̂
[
( )] ̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=
̂ ̂
(
(
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂
Let ⃗⃗⃗ =⃗⃗⃗⃗⃗⃗ = ( Then ̂ = |⃗⃗⃗ | =
)
Then
= ⃗⃗⃗
P(
(
)
Solution: Given
At
)
]
Q#07: Evaluate the directional derivative of
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
(
]
=[
has coordinates (
)
)
( )
=
̂ √
( ̂
̂
)̂
)̂
(
)̂ [ ( )] ̂
̂ ̂
)
(
(
=
) =( ̂
̂
) ̂
(
)̂
)̂= ̂
(
̂
̂
̂
√
Thus Directional derivative of
at Point P in the of ⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =( ̂
̂
̂ ̂).
̂
̂ √
̂
=
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
√
=
√
Page 17
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
at P in the direction of ⃗⃗⃗ where
Q#08: Find the directional derivative of
(
(i)
) and (
(ii) (
(iii) ()
(
Solution: Given ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂ ̂
P(
At
Let ⃗⃗⃗ = ̂
)
and
⃗⃗⃗ =
̂
and
⃗⃗⃗ =
̂ ̂
̂ ̂
̂
Then
= ⃗⃗⃗ =
)
and ⃗⃗⃗ = ̂
)
̂
⃗⃗⃗ = ̂
̂
̂=
(
̂
̂
) ̂
(
)̂
)̂
(
̂ ̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
) ̂
=
̂ ̂ ̂
⃗⃗⃗
̂ = |⃗⃗⃗ | =
Then
√( )
( )
(
)
̂ ̂
=
̂ ̂
=
√
√
Thus at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Directional derivative of ( )
(
⃗⃗⃗ =
) and
Solution: Given ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
= ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=(
At
P(
̂
̂ ̂
̂) . ̂
=
√
√
=
̂
Then ̂
) ̂
̂
̂ =(
(
) Let ⃗⃗⃗ =
̂= ̂
)̂
(
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ̂
(
(
)̂
)̂
(
)̂ = [ ( )] ̂
̂
) ̂
[ ( )] ̂
Then
[ ( )] ̂ = ̂
̂
√( )
(
⃗⃗⃗
̂ = |⃗⃗⃗ | =
̂
̂ ̂ )
( )
=
̂
̂
=
√
̂
̂
√
Thus Directional derivative of
at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =( ̂ =
̂
̂ ̂).
̂
̂
√
=
√
√
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 18
√
Vector Analysis: Gradient, Divergence and Curl ( )
) and ⃗⃗⃗ =
(
̂
= ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=(
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=
̂
̂= ̂
) ̂
(
̂
) ̂
(
)̂
)̂
(
)̂
(
̂] ̂
( ) ( ) ( )
=
̂ ̂
( )̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
)
Let ⃗⃗⃗ = Thus
Then
[ ̂
P(
At
̂ ̂
Solution: Given ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
B.Sc & BS Mathematics
[ ̂
̂]= ̂
[ ̂
⃗⃗⃗
̂ = |⃗⃗⃗ | =
Then
̂ √(
̂ ̂
)
̂] ̂
( )
̂
=
( )
at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Directional derivative of
[ ̂
=
̂
=
̂ ̂
√
̂ ̂
̂ ]. ̂
[
=
̂ ̂
√
̂
̂
√
] √
=
√
Q#09: Find the directional derivative of ()
(
( )
(
) in the direction of
) in the direction of
( )
(
̂
(
=(
) ̂
At
P(
= ̂
̂
Thus
)̂ ) ̂
̂
) in the direction of
̂ ̂
Then ̂
(
̂ ̂
Solution: Given = ⃗⃗⃗
̂ ̂
) along z-axis.
(i)
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂
̂= ̂
(
(
) ̂
(
)̂
)̂
(
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
)̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ Then
= [(
) ] ̂ ⃗⃗⃗
̂ = |⃗⃗⃗ | =
Directional derivative of
[ ( )( ̂
√( )
̂ ( )
( ) ] ̂
)
̂ ( )
=
̂
̂
)( )] ̂ =
[ (
̂
√
=
̂
̂ ̂
√
=
at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =(
̂
̂
̂) .
̂
̂
̂
̂
̂
̂
Let ⃗⃗⃗
̂
̂
̂ ̂
=
=
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
= Page 19
Vector Analysis: Gradient, Divergence and Curl (
(ii)
= ⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=
̂
(
) ̂
(
)̂
)̂
(
̂ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
= [( )( )] ̂
̂ ̂
̂= ̂
̂
)
Let ⃗⃗⃗ = ̂
̂ ̂
Then
̂
P(
At
̂
) in the direction of
Solution: Given ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
B.Sc & BS Mathematics
⃗⃗⃗
̂ = |⃗⃗⃗ | =
Then
Thus Directional derivative of
[( )( )] ̂ = ̂
[( )( )] ̂
̂ ̂ ̂
√( )
( )
( )
=
̂ ̂ ̂
√
(
= ⃗⃗⃗
=( P(
At ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
√
=
√
̂ ̂ ̂
̂ =( ̂
√
=
√
̂) . ̂
̂
̂ ̂ √
=√
Then ̂
) ̂
̂ ̂ ̂
) along z-axis.
Solution: Given ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
=
at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =
(iii)
̂ ̂
̂= ̂
(
)̂
(
) ̂
(
)̂
)̂
(
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
)̂
(
)
=[ ( ) =[
)( ) ] ̂
( ] ̂
[
] ̂
Let ⃗⃗⃗ = ̂ (along z-axis)
[
[
( )( ) ] ̂ ] ̂= Then
[ ̂
̂
( )( )
( )(
)( )] ̂
̂
̂=̂
Thus Directional derivative of
at Point P in the direction of ⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂ =(
̂
̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
̂ ) . ̂ = 36
Page 20
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
Q#10: Prove that (i) ⃗⃗⃗
⃗⃗⃗
=n ⃗⃗⃗
(i)
(
(ii)
(
(ii)
⃗
[
⃗⃗⃗
=[
⃗⃗⃗
=
[
⃗⃗⃗
=
[
⃗⃗⃗
=
̂
̂ ] ̂
̂
̂]
=
[ (
=
[
=
[
)
̂ ̂ ] ̂
[
̂] ̂
̂] ̂
()
Hence proved.
⃗⃗⃗
=
then ](
]
̂ ] ̂
̂
)=[ =
(
(iii)
̂ ------(i)
[
̂
⃗⃗⃗
) =
=[
̂
⃗⃗⃗
Solution: We know that (
⃗⃗⃗
⃗⃗⃗
=n
Solution: We know that ⃗⃗⃗ Then
⃗⃗⃗
)=
)]
[
(
] ]
[
)
(
)]
[
[ [
]
(
(
[
]
)
(
)
)]
]
]
[
)
[ [
] ]
]
[
]
[
[ ]
]
[
]
[
]
= [
]
[
]
[ ) =
⃗⃗⃗
]
[ [(
] (
[
⃗⃗⃗
̂
]
[
̂
̂) ( ̂
]
[ ̂
] ̂) ]
[
]
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 21
Vector Analysis: Gradient, Divergence and Curl (
(iii)
)
Solution: We know that
=
[
then
] [
=
=n {
]
[
(
)
(
]
[
]
)
(
)}
}
= [(
)
( )
= [(
)
{( )
= [(
)
{( )
⃗
̂
( )
)
)
]
[
]
) ( )
}] ]
}]
{
} ] ---------------(a)
then
( ) Differentiate w. r. t x
[
{(
(
{
() } ̂
} ( )
( ) }
( )
]
)
{( (
̂
[
=
)
= [{(
Let
B.Sc & BS Mathematics
------(i) 2r
= 2x
Similarly
( )
Again differentiate w. r. t x
=
( )
=
=
Similarly = [(
)
{
= [(
)
{
= [(
)
{ }
= [(
)
( )
= [(
)
= [(
)
(
)
}
{
}
}]
{ {
(
{
}] )
}]
}]
{ }] ]= [(
)
]
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 22
Vector Analysis: Gradient, Divergence and Curl Q#11: Prove that (i) ⃗⃗⃗ ⃗⃗⃗
(i) Solution:
⃗
=2
= 3r ⃗ ⃗
Let
⃗
̂
̂ ̂
[
̂] ̂ ] ̂
] ̂
=[
̂ ] ̂
] ̂
[
] ̂
[
()
] ̂
[
̂ ̂
{
}
̂] ̂
[ ̂
=
------(i)
=
[
̂
=[
then
̂ ̂
=[
⃗⃗⃗
̂
̂
We know that ⃗⃗⃗ Then
(ii) ⃗⃗⃗
= 3r ⃗
B.Sc & BS Mathematics
̂] ̂
()
= 3r ⃗
Hence proved.
⃗⃗⃗
(ii) Solution:
⃗
Let
̂
We know that ⃗⃗⃗ Then ⃗
⃗
=2
[
̂ ̂
] ̂
[ ̂
= =2
⃗
] ̂
]̂
[
] ̂
[
̂ ̂
] ̂
() {
}
̂] ̂
̂
------(i)
̂
=
[
[
̂
=[
⃗⃗⃗
̂] ] ̂
=[
then
̂ ̂
̂
[
̂ ̂
̂] ()
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 23
Vector Analysis: Gradient, Divergence and Curl Q#12: Prove that ⃗⃗⃗
(i) Solution:
⃗⃗⃗
(i)
B.Sc & BS Mathematics ⃗⃗
⃗⃗⃗ ( ) =
= ̂ (ii)
=̂ ⃗
Let
̂
We know that ⃗⃗⃗ ̂ ⃗
Then
̂
̂] ̂
̂
=
̂
------(i)
̂
=
̂ ̂
=
̂ ̂
()
̂ ̂
=
then
̂ ̂
[
⃗⃗⃗
̂ ̂
{
}
⃗⃗
=̂
̂ =
⃗⃗
Hence proved. ⃗⃗
⃗⃗⃗ ( ) =
( ) Solution:
⃗
Let
̂
We know that ⃗⃗⃗ Then ⃗ ( )
[
̂
=(
)
̂
=
( ̂
[ ⃗⃗
̂
)=
̂
]=
(
)
̂
̂
( (
)̂
(
)̂
̂
) ()
̂
[
) ̂ ̂
̂
̂
------(i)
̂
̂] (
̂
=
then
̂
̂
=
⃗⃗⃗
̂ ̂
{
̂
}
]
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 24
Vector Analysis: Gradient, Divergence and Curl
⃗⃗⃗ (⃗⃗ ⃗ ) = ⃗⃗ where ⃗ is a position vector.
Q#13: Let ⃗⃗ be a constant vector show that Solution: Let ⃗⃗ Then
̂
⃗⃗ ⃗ = (
̂
̂
We know that ⃗⃗⃗ Then ⃗ (⃗⃗ ⃗ )
̂
&
̂) ( ̂
⃗ (⃗⃗ ⃗ ) =
) ̂
̂
⃗
Solution: Given that
̂
[
(⃗⃗ ⃗ ) ̂
(⃗⃗ ⃗ ) ̂
(⃗⃗ ⃗ ) ̂ )̂
)̂
(
̂
( ) ̂
̂ ̂
̂
]
then
------(i)
̂] ( )
⃗⃗⃗ ̂
̂
̂
( )̂ ̂ ̂]
̂
̂ ̂
̂
( )̂
= ( )
⃗⃗
̂) =
(
̂
̂
( ) ̂
= ( )[
̂
̂
Then ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ f(r) = ⃗ ( )
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ f(r) =
̂
Hence proved.
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ f(r) where ⃗ =
=[
̂
̂ ̂
⃗ (⃗⃗ ⃗ ) =⃗⃗⃗ Q#14: Find
̂
̂
̂ ] (⃗⃗ ⃗ )= ̂
(
=
⃗
̂ ̂
̂
[
̂ ̂
B.Sc & BS Mathematics
( )
̂ ()
{
}
( )
( )
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 25
Vector Analysis: Gradient, Divergence and Curl ⃗⃗
Q#15: If
̂
⃗⃗⃗
̂
⃗
⃗⃗
⃗
⃗⃗
⃗
̂
=
) ̂
= ⃗⃗
Now
̂] ̂
(
=
̂
⃗
⃗⃗
(
)
̂ ̂
̂ ̂
̂
[
⃗
⃗⃗
⃗⃗
̂
⃗
Then
̂ ̂
Solution: : Given that If We know that ⃗⃗⃗
B.Sc & BS Mathematics
(
̂ ̂
)̂
)̂
(
̂ ̂
̂) (
(
̂
̂
(
)(
)
( ) (
)
(
̂
)(
)
̂) ̂
(
)( )
)
At
̂ Now ⃗⃗
⃗
|
⃗⃗
⃗
= ̂(
( ) (
)( )
( )( ) =
–
̂ ̂
|= ̂ |
)
|
̂(
)
̂|
|
̂(
̂|
|
)
)
At ⃗⃗
⃗
= ̂[
(
= ̂[
)( )
⃗⃗
⃗ ̂
̂
]
̂[
( ) ( ) ] ]
̂[
̂[ ( )
( ) (
)( ) ]
̂[
( )
( ) (
) ( )]
]
̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 26
Vector Analysis: Gradient, Divergence and Curl ⃗
⃗
Q#16: If
Let ⃗
Solution: Given that We know that ⃗⃗⃗ ⃗
Then
⃗⃗⃗
=
⃗⃗⃗
=[
̂] (
̂
̂
] ̂ ̂
[
( ]̂
)̂
] ̂
[
=
]̂
⃗
̂
[
⃗⃗⃗
=
(
⃗⃗⃗
=
̂
̂] ( ̂ ) ̂
̂
⃗
=[
̂
̂
(
) ̂
(
)̂
(
̂
̂
)̂
)
(
)̂
)̂
(
)̂
(
Now taking dot product of ⃗ ⃗
)=
̂
̂ ̂
̂
[
̂] ( ̂
̂
=
⃗
⃗ )
We know that ⃗⃗⃗
Solution: Given that
&
]
⃗⃗⃗ (⃗
⃗⃗⃗
] ̂)
[
Hence proved.
Q#17: If
Then
)̂
(
=n[
⃗
⃗
------(i)
]̂
[
̂ ) ([ ̂
̂ then ̂
)
) ̂
(
̂
̂ ̂
(
⃗
⃗
Now
̂ [
B.Sc & BS Mathematics
⃗ . ̂] [ ̂
̂
) ̂ ]= 6xy
(
Now applying ⃗⃗⃗ operator ⃗⃗⃗ (⃗
⃗ )
⃗⃗⃗ (
) (
⃗⃗⃗ (⃗
⃗ )=(
[ ) ̂
) ̂
(
)̂
̂
̂
̂] (
(
) ̂
(
)̂
) (
)̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 27
Vector Analysis: Gradient, Divergence and Curl ⃗ ( )
Q#18:
Then
⃗
⃗
Solution: Let
̂
⃗ ( )
̂ ̂
̂
[
B.Sc & BS Mathematics
( ) ̂ ( )
=
( )[
⃗⃗⃗
( ) ̂
̂
̂
̂ ̂
̂ ̂
̂
( )
()
̂] ̂
̂
̂
( )̂
( )̂
( )[
------(i)
̂] ( ) ̂
=
then
{
}
]
⃗⃗
⃗ ( ) = ( )
Now taking cross product with ⃗ ⃗ ( )
⃗ =
⃗ ( )
⃗ =0
⃗⃗
( )
(⃗⃗ ⃗ )⃗
Now
Let ⃗⃗
(
⃗
)
⃗
(⃗⃗ ⃗ )⃗
⃗⃗
̂
⃗⃗ ⃗ = (
̂ ̂
̂
̂
(⃗⃗ ⃗ )⃗ =
̂ ](
̂
( ̂
̂
̂) ̂
̂
̂
̂) =
̂ ̂
̂
̂) ( ̂
⃗
&
̂
[ =
(⃗⃗ ⃗ )⃗
( )
⃗ )=
⃗⃗
We know that ⃗⃗⃗ Then
(⃗
Hence proved.
Q#19: Solution:
( )
⃗ ==
̂
̂) ̂
(
̂
̂
̂)
(
̂
̂
̂)
̂
⃗⃗
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 28
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
Divergence of a Vector: Le ⃗⃗ (
) is a vector .Then Divergence of a vector ⃗⃗ is defined as;
Div ⃗⃗ = ⃗
⃗⃗
.
Solenoid Vector: A vector ⃗⃗ is said to be Solenoid, if
Div ⃗⃗ =
.
Properties of the Divergence: If ⃗⃗ & ⃗⃗⃗ are two vector & (i)
Div (⃗⃗
is a scalar function then
⃗⃗⃗ ) = ⃗ (⃗⃗
⃗⃗⃗ ) = ⃗ ⃗⃗
(ii) Div ( ⃗⃗ ) = ⃗ ( ⃗⃗ ) =
⃗ ⃗⃗⃗
( ⃗ ⃗⃗ ) (⃗
) ⃗⃗
Curl of a Vector: Le ⃗⃗ (
) is a vector .Then Curl of a vector ⃗⃗ is defined as;
Curl ⃗⃗ = ⃗
⃗⃗ .
Irrotational Vector: A vector ⃗⃗ is said to be Irrotational, if Curl ⃗⃗ =
.
Properties of the Curl: If ⃗⃗ & ⃗⃗⃗ are two vector & (i)
Curl (⃗⃗
is a scalar function then
⃗⃗⃗ ) = ⃗
(ii) Curl ( ⃗⃗ ) = ⃗ (iii) Curl (⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ) (iv) Curl (
(⃗
( ⃗⃗ ) = Curl (⃗
(⃗
⃗⃗⃗
⃗
⃗⃗ ) (⃗
⃗
)
Curl (⃗ ⃗⃗ )
⃗⃗
(⃗
⃗
)
)
0
(⃗ ⃗⃗ )
0
⃗⃗
⃗⃗⃗ are two vector functions. Then prove that
Theorems: If ⃗⃗ ( )⃗
⃗⃗ )
⃗⃗⃗ ) = ⃗
(⃗⃗
⃗⃗ )
(⃗ ⃗⃗ ) ⃗ ⃗
(⃗
⃗⃗
⃗⃗ )
(⃗ ⃗⃗ ) ⃗
⃗⃗
(⃗
⃗)
(⃗ ⃗ ) ⃗
⃗
(⃗
⃗⃗ )
(⃗ ⃗⃗ ) ⃗
(⃗ ⃗⃗⃗ )⃗⃗
⃗
(⃗
⃗⃗ )
(⃗ ⃗⃗ ) ⃗
⃗⃗
Proof: We know that ⃗ Then
(⃗ ⃗⃗⃗ )
Here ⃗ ⃗⃗⃗
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 29
Vector Analysis: Gradient, Divergence and Curl Example#01: Find the divergence of ⃗⃗ ̂
Solution: Given ⃗⃗
(
Div ⃗⃗
=
=
=
̂
(
)
(
(
[
(
) )
[ ]
(
)
(
̂
(
)
(
)
) ( )
(
(
)
(
) (
)
) ( )
(
)
) [
]
(
)
)
(
(
]
) )
=(
)
)
(
=(
)
) (
) [
̂
̂
)
(
) ̂
)
)
(
(
̂
̂
(
) ( )
(
̂ ̂
⃗ ⃗⃗
̂ ) ( ( ̂
̂
⃗⃗
=(
)
We know that =(
̂ ̂
B.Sc & BS Mathematics
(
(
) [
)
(
) (
(
)
)
] )
]
Hence Div ⃗⃗
0
Example#02: If ⃗⃗ = ̂
(i) (⃗⃗ ⃗ )
Find
Solution: Given ⃗⃗ =
̂ ; ⃗⃗⃗ ̂
(ii) (⃗⃗
̂
⃗ )
̂
(ii) ⃗⃗
⃗
̂ ; ⃗⃗⃗ ̂
̂ and ̂
̂
(iv) (⃗⃗⃗
⃗⃗⃗
⃗⃗ )
̂ and ̂
(i) (⃗⃗⃗ ⃗ ) (⃗⃗ ⃗ )
̂
= [(
̂) ( ̂
=( (
(⃗⃗ ⃗ )
=
(
) )
)
)
)
( (
̂ )]( ̂
)(
= =
̂
) (
(
)
)
)
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 30
Vector Analysis: Gradient, Divergence and Curl (ii) (⃗⃗⃗
⃗ ) ̂
(⃗⃗
⃗ )
̂ ̂
=|
̂[
]
= ̂[ (
̂[
(
= ̂[
)
(
(⃗⃗
)] (
)
(
̂
[ ̂
=[
̂ ̂] ̂
̂
=[
̂
| (
)
(
]
̂[
̂]
[
)
(
(
)]
̂[
)] ]
(
)
̂[
(
)]
]
̂|
|
)̂
(
)̂]
̂
̂
̂
̂ ]=|
|
̂|
(
̂[
|]
) ]
(
)]
̂[
̂[
(
)
(
)]
]
⃗⃗⃗ ̂
⃗⃗ )
(
̂
̂[
̂] ̂
) ̂
)]
] ⃗⃗⃗ )
(
̂
[
(
= ̂[
(⃗⃗⃗
̂[
[
̂]
[ ̂|
( )(⃗⃗⃗
)
⃗
⃗
= ̂[
]
)]
)]
⃗ ) = ̂[
( )⃗⃗⃗
̂[ (
̂[
|]
]
]
(
̂|
|
̂[
]
)
̂|
|
̂[
]
= ̂[
=[ ̂ |
|
= ̂[
⃗⃗
B.Sc & BS Mathematics
̂ ̂
⃗⃗⃗ = |
|
={̂ [ ={ ̂[
] ]
̂[
⃗⃗⃗ = [ ̂ |
̂[ ]
|
̂[
] ̂[
]}
̂|
̂|
|
]}
|]
⃗⃗⃗
⃗⃗⃗
⃗⃗⃗
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 31
Vector Analysis: Gradient, Divergence and Curl ̂
={
[
]̂
̂
B.Sc & BS Mathematics
] ̂} {
[
̂
̂ ̂
|=[ ̂ |
=|
̂ [(
={
)
|– ̂|
) ] ̂[ ( ) (
(
={ ̂[
|]
) ] }
̂[
]
̂|
|
̂[ ( ) ( ) ]
]
̂[ Example#03: If
̂} ̂
]
̂[
̂[
]}
̂[
]
]
, Find Div(⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )
2
Solution: We know that ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗ ̂
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
̂ = ̂
̂
(
) ̂
(
)̂
)̂
(
̂ ̂
Now Div(⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )
⃗ ⃗ =
=( (
̂
̂
)
(
̂) ( )
̂
̂) ̂
(
)
Div(⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ) = Example#04:Show that Div Solution: Let ⃗ Now Div =(
̂
̂ Then ̂
⃗ =⃗ ( ̂
⃗ = 10
⃗ )
( ̂) (
̂
=[ ( )
]
̂
r = |⃗ | = √ ̂) ( ̂
̂
̂
[
]
(
)
(
[ ( )
=[
=
------------(i)
̂] ) ̂
̂)= ̂
[ ( )
or
)
(
)
]
]
=[
(
=[
(
)]
)]
() {
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
} Page 32
Vector Analysis: Gradient, Divergence and Curl =[
(
=[
(
=[
)] )] ]
=[ Div
B.Sc & BS Mathematics
] ⃗ =
Hence proved. Example#05: Show that Div Solution: Let ⃗
̂
⃗⃗
= 0. ̂ Then
̂
r = |⃗ | = √
or
=
----------(i)
Now Div
⃗⃗
⃗ =⃗ (
= Div =( = =[
̂ (
(
̂
̂) ( ̂
) ( )
⃗ )
(
̂
)
(
)
̂
̂] )
[ ( )
(
)
]
[ ( )
(
)
]
]
=[
(
=[
(
=[
(
=[
(
=[
⃗⃗
̂
)
=[
Div
[
̂) ̂
( ]
̂) ( ̂
)] )] )] )] ]
=[
]
= 0
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 33
Vector Analysis: Gradient, Divergence and Curl Example#06: If ⃗⃗
̂
̂ . Show that Curl( ̂ ̂
⃗⃗⃗
⃗⃗
Solution:
| =[ ̂ |
(
= ̂[
)]
] ̂
⃗
⃗⃗ )
Curl(
) ̂[
̂ [ ]= 0 ̂
]
̂ ̂
̂|
)
(
|] ̂[
)]
(
)]
̂
]
⃗⃗
⃗
̂
]̂
̂[ [
=[ ̂ |
|
̂|
̂[
] ]̂
̂ ⃗⃗ = ⃗⃗⃗
̂ ̂
| =[ ̂ |
]
⃗⃗
̂ ̂
|
= ̂[ ⃗⃗ = [
⃗⃗
& ⃗
̂
Now
̂[
⃗ then prove that ⃗⃗ ̂
⃗⃗
(
̂[
]
|
Hence proved.
⃗⃗
Solution: Let ⃗⃗
Then
̂|
⃗⃗ ) = 3 ̂
Example#08: If ⃗⃗
]
̂ ̂
|
= ̂[ ]
)
|]
|
=[ ̂ | (
(
̂|
]
⃗⃗ =|
= ̂[
|
]̂
̂ Now Curl(
̂[
[ ̂
̂[
]
]
̂|
|
̂[
̂[
)]
(
⃗⃗ = [
⃗⃗ ) = 3 ̂ .
̂ ̂
⃗⃗ = |
= ̂[
B.Sc & BS Mathematics
[
̂|
|
|]
] ]̂ ̂
̂
⃗⃗ = |
|
|
̂|
|
̂|
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
|]
Page 34
Vector Analysis: Gradient, Divergence and Curl = ̂[
(
)
̂[ (
=2 ⃗⃗
= 2(
(
)
= ̂[
]
̂
̂[
(
)
(
)]
)] ]
̂[
]
̂ ̂
̂
)]
(
̂[
̂
B.Sc & BS Mathematics
̂)
⃗⃗ = 2 ⃗⃗ ⃗⃗
⃗⃗
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 35
Vector Analysis: Gradient, Divergence and Curl
B.Sc & BS Mathematics
Exercise# 4.2 Q#01: Fin the divergence & curl of the vector functions. (i) ⃗⃗ = (
(
) ̂ ⃗⃗ = (
(i)
)̂ (
) ̂
) ̂ (ii) ⃗⃗ =(
( )̂
) ̂
(
)̂
)̂
(
)̂
(
Solution: We know that Div ⃗⃗
⃗ ⃗⃗ = ( (
=
̂
̂
)
(
̂ ) [(
(
) ̂
)
(
)̂
) ̂]
(
)
Div ⃗⃗ = 2x +2y+2z ̂
⃗ = ⃗⃗⃗
̂ ̂
⃗⃗ = |
| =[ ̂ |
(
= ̂[ = ̂[
) ]
⃗ =
̂[
̂
( ) ⃗⃗⃗ =(
( ]
̂[
(
̂[
)
̂|
(
̂|
|
)]
̂[
(
)
(
)
|]
(
)]
]
̂ ̂
) ̂
)]
|
(
)̂
)̂
(
Solution: We know that Div ⃗⃗
⃗ ⃗⃗ = ( ̂
̂ ) [( ̂
) ̂
(
)̂
) ̂ ]=
(
(
)
|
̂|
(
)
Div ⃗⃗ = ̂ &
⃗ = ⃗⃗⃗
⃗⃗ = |
(
= ̂[ = ̂[ ⃗ =
̂ ̂
̂
( ̂
| =[ ̂ |
)
(
)]
̂ [(
)] )
(
̂[ ]
|
̂[
) (
(
̂|
)]
̂[
(
)
|]
(
)]
)]
̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 36
Vector Analysis: Gradient, Divergence and Curl Q#02: Find Div ⃗ &
⃗ where
(i) ⃗⃗ = grad(
(ii) ⃗⃗ =(
)
(i) ⃗⃗⃗ = grad(
) ̂
(
)̂
(
)̂
) ⃗⃗ = ⃗⃗⃗ (
Solution: Given ⃗⃗ =
B.Sc & BS Mathematics
)=(
(
) ̂
⃗⃗ = (
) ̂
̂
(
(
̂ )( ̂ ) ̂
) ̂
) )̂
(
)̂
(
We know that Div ⃗⃗
⃗ ⃗⃗ = ( ̂
(
=
̂ ) [( ̂ (
)
(
) ̂
)
(
)̂
)̂]
(
)
Div ⃗⃗ = ̂ ⃗ = ⃗⃗⃗
&
⃗⃗ = |
|
= [ ̂|
= ̂[
= ̂[ = ̂[
̂ ̂
|
(
(
)
(
)]
̂[
] ⃗ =
̂|
|
)]
̂[
(
̂
̂
)] ̂[
]
̂[
(
)
̂[
(
̂|
(
|]
)]
̂[ (
)
(
)]
)]
]
̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 37
Vector Analysis: Gradient, Divergence and Curl ( )⃗⃗⃗⃗ = ̂
B.Sc & BS Mathematics
̂ ̂
Solution: We know that Div ⃗⃗
⃗ ⃗⃗ = ( ̂
̂ ) [ ̂
̂
̂ ]= ̂
(
)
(
)
(
)
Div ⃗⃗ = ̂ ⃗ = ⃗⃗⃗
̂ ̂
⃗⃗ = |
= ̂[
(
|=[ ̂ |
)
= ̂[
( ]
⃗ =(
)] ̂[
) ̂
(
̂[
̂[
)]
(
) ̂
(
)
̂
(
)
)
( ̂
⃗ =0
)
̂ [ ((
)
⃗⃗⃗
⃗⃗ = 0
̂ has its curl equal to zero .
̂ ̂
|=0
|
)
[̂
̂[
)
((
]
̂ [( ̂
[(
)
(
)
( )
(
)
(
)
̂|
|]=0
) ) )]
)]
|
̂|
|
((
(
̂
( [ ̂| (
)
)
̂ Given condition :
(
|]
)̂
Q#03: Find m , so that the vector( ) ̂
(
̂|
|
]
( ̂
Solution: Let ⃗ = (
)
̂[
]
̂|
|
)]
̂ [ ((
)]
̂[ (
] ̂
[ (
) )
) )
(
)] = 0
] ]̂=0
Putting coefficients of ̂ is equal to zero. (
)
–
–
By using cancelation Property
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 38
Vector Analysis: Gradient, Divergence and Curl ⃗ =n
Q#04: (i) Show that Div (i)
̂
=(
̂
⃗ )
)
(
( )
=[
(
r = |⃗ | = √
̂ ̂
)
(
]
=[
(
)
=[
(
)
(
=[
(
)
(
=[
(
)
(
=[
(
)
(
= [
(
)
̂
[
=
------------(i)
̂] ) ̂
̂) ̂
(
)
=[
)
or
̂) ( ̂
̂) ( ̂
(
=
) ( )
[ (
(
)
]
)
( )
[
(
)
(
)
]
]
() )]
{
}
)] )] ) ] ]
]
= [( Div
̂ Then ̂
⃗ =⃗ (
Now Div
(
(ii) show that
⃗ =n
Show that Div
Solution: Let ⃗
B.Sc & BS Mathematics
]
)
⃗ =
Hence proved.
( )
(
)
Solution: We know that
=
then
[
]
=
[
]
=
[(
)
[
] ]
[(
[
] )
]
[(
)
]
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 39
Vector Analysis: Gradient, Divergence and Curl =(
){ (
=(
) [{(
)
(
=(
}
)
( )
)
{(
}
)
(
( )
]
) [(
)
{( )
=(
) [(
)
{( )
⃗
)}
(
( )
=(
Let
(
}] ) [(
)
)
)
)
{(
B.Sc & BS Mathematics
̂
̂ ̂
( ) Differentiate w. r. t x
( )
( ) }
( )
{
() }
{
then 2r
}] } ] ---------------(a)
------(i)
= 2x
Similarly
( )
Again differentiate w. r. t x
( )
=
=
=
Similarly Putting values in Equation (a) =(
) [(
)
{
=(
) [(
)
{
=(
) [(
)
{ }
=(
) [(
=(
)[( (
}]
{ {
( ) )
)
{
}
)
) [(
=(
}
(
}] )
}]
{
}]
{ }] ]= (
)[(
)
]
)
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 40
Vector Analysis: Gradient, Divergence and Curl Q#05: If ⃗ =
̂
⃗
⃗ ⃗ At (
̂
)(
)
⃗ ⃗
) ⃗
( )
(
̂
)(
)
)̂
)̂
(
̂ ) ̂
(
)( )
⃗
⃗
]
̂[
]
̂[ (
̂ ) ̂
] ̂ [(
)̂
. Show that
(
)̂
̂[
]
|
̂|
(
)
(
̂[
( )
|]
)(
)]
] )( ) ]
( ) (
⃗
⃗
̂
=
(( ) (
) ( )]
̂ ̂
⃗ curl ⃗
̂ |= [ ̂ |
( )]
̂[
)]
[̂ ( )
]
̂
))
= ̂[ ̂
̂
̂|
̂[
( ) ( ) ]
⃗⃗ =|
( (
( )( )
( ]
)( )
̂
= ̂[
( )
̂[ (
Solution: Given ⃗ = (
⃗
)( )
|
̂[
= ̂[
) ̂
Now Curl ⃗
( ) (
)
)]
]
Q#06: If ⃗ = (
Now ⃗ curl ⃗
(
|=[ ̂ |
= ̂[
⃗ =
).
̂
= ̂[
At ( ⃗
(
= |
⃗
) ̂
( ) (
= ̂
= ̂[ ⃗
(
at point (
then
̂) (
̂
̂ ( )⃗
&
⃗
=
)
⃗
(ii) ⃗
̂ ̂
=(
̂
̂=
̂ =(
. Find (i) ⃗ ⃗
̂ ̂
=
⃗ ⃗
(i)
̂
̂
⃗
̂ & ̂
Solution: Given ⃗ =
B.Sc & BS Mathematics
(
)
̂[
( (
̂[
| )
(
))
̂|
(
(
)]
̂] = (
)(
̂[
)
( )
|
̂|
|]
(
)]
]
̂ ) ̂ ̂
(
)̂] [ ̂
̂
)
( )( )
(
)(
)
= ⃗ curl ⃗⃗ = 0 Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 41
Vector Analysis: Gradient, Divergence and Curl ̂
Q#07: If ⃗
⃗ ⃗⃗⃗ =
̂
̂ ̂
=
√
̂
⃗⃗⃗
√
√
̂
√
(
(
)
(√
√
̂
√
( )
̂ ̂
̂
̂ ) ( ̂
√
(ii) ⃗
√
√ ̂
Solution: Given ⃗
=
. Show that (i) ⃗ ⃗⃗⃗ =
√
(i)
⃗ ⃗⃗⃗ = (
̂ ̂
B.Sc & BS Mathematics
( )
) √
+
)
(
√
(
)
(√
√
)=
√
)
√
( )
√
)
√
√
(
(√
)
(
)
)
√
√
√
√
= [
]
[
√
]
[
√
] √
= = = ⃗ ⃗⃗⃗ =
(
)√
(
)√
(
=
⃗
̂ √
= = ⃗
√
⃗⃗⃗
(
̂
) )√
√
√
√
̂ ̂ |= | √
√
( )
{ ̂[
]
̂ ̂
|
|
̂[ ( )
( )]}
{
√
}
√
|
{ ̂[
̂ ̂
| |
[ ̂|
√
=
̂
√
√
̂ ̂
̂
=
(
=
)√
)√
⃗⃗⃗
Solution: Given ⃗
⃗⃗⃗
(
(
Hence proved .
√
(ii)
⃗
)√
̂|
|
( )] ̂[
|]
( )
̂[ ]
̂|
̂[
( )] ]} =
√
{ ̂
̂
̂ }=
√
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 42
Vector Analysis: Gradient, Divergence and Curl ̂
Q#08:If
̂ Then show that Div( ⃗ ) ̂ ̂
Solution: Given ⃗ ( ⃗ )
Div ( ⃗ )
⃗ [(
̂
̂
̂ ) [ (
(
̂
̂
̂ ) [(
(
)
̂
5
̂ ̂
)(
(
B.Sc & BS Mathematics
̂ )] ̂
) ̂
(
) ̂ (
)̂
(
)
) ̂]
( )̂
(
(
) ̂]
)
= Div( ⃗ )
5(
Div( ⃗ )
5
) Hence proved.
Q#09:If ⃗ is a constant vector and (i) ⃗ (⃗ ⃗ )
(ii) ⃗ (⃗
⃗
⃗ ⃗
Then
̂
( ̂
&
⃗
̂
[(⃗ ⃗ )⃗ ] = ⃗
̂ & Let ⃗ ̂
̂ )( ̂
̂ .Show that ̂
( )
⃗ )
̂
Solution: Given
̂
̂
|= ̂ |
̂) = ̂
̂|
|
̂|
|
⃗ = ̂[
]
̂[
]
̂[
]
⃗
⃗ = ̂[
]
̂[
]
̂[
]
Let ⃗ (⃗ ⃗ )
(
⃗ (⃗ ⃗ )
̂
̂
̂ )(
⃗
) )
̂
=
|
⃗
(
=
[(⃗ ⃗ )⃗ ] = 4(⃗ ⃗ )
̂ ̂
⃗
⃗ ( ⃗ ⃗⃗ )
( )
̂ ̂
⃗ =|
(i)
⃗
̂
̂
(
) ̂
(
) ̂
̂ Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 43
Vector Analysis: Gradient, Divergence and Curl ⃗ (⃗
( )
⃗ (⃗
Let
⃗⃗ ) ⃗ )
(
̂
̂ ) ( ̂[ ̂
[
⃗ (⃗ ( )
]
]
̂[
̂[
]
[
])
]
Hence proved
[( ⃗ ⃗⃗ )⃗⃗ ] = ⃗
(⃗ ⃗ )⃗
]
[
⃗ ) ⃗⃗
(
)(
( (⃗ ⃗ )⃗
B.Sc & BS Mathematics
̂
̂
) ̂
(
) ̂
̂)
(
)̂
(
)̂
( )̂
)̂
(
Now ̂ ⃗
[(⃗ ⃗ )⃗ ]
[(⃗ ⃗ )⃗ ]
|
| (
)
= ̂| (
= ̂[
)
(
(
)
|– ̂| ( )
) ( ̂| (
(
)
(
)]
̂[
(
)
(
)]
[(⃗ ⃗ )⃗ ] = ⃗
̂[
] ⃗
(
)
| )
)]
(
]
(
| )
(
)
)
̂[
= ̂[
̂ ̂
̂[
]
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 44
Vector Analysis: Gradient, Divergence and Curl ( ) Let
B.Sc & BS Mathematics
[( ⃗ ⃗⃗ )⃗⃗ ] = 4( ⃗ ⃗⃗ ) (⃗ ⃗ )⃗
(
)(
( (⃗ ⃗ )⃗
) ̂
(
̂
̂) ̂
(
) ̂
)̂
(
)̂
( )̂
)̂
(
Now [(⃗ ⃗ )⃗ ] = ⃗⃗⃗ =(
=
̂
[(⃗ ⃗ )⃗ ] ̂ ) [(
̂
) ̂
(
)
(
)̂
(
)
) ̂]
(
(
)
= = (
)
[(⃗ ⃗ )⃗ ] = 4(⃗ ⃗ ) Q#10:If ⃗
Hence proved.
( ) ̂ ̂
Solution: Given ⃗
( ( ) ̂
̂ ̂
Then Curl ⃗
⃗⃗⃗
⃗
) ̂ then evaluate Curl ⃗ ) ̂
( ̂
̂
|
|= ̂ | ( )
(
= ̂[ = ̂[
) (
)
=
(
) ̂
Curl ⃗ =
(
) ̂
[
(
)
( )]
̂[
]
̂[
]
̂
[ ( )
| )
̂|
)
]
̂[
( )
]
( )
( ̂[
(
(
)
|
( )
̂|
( )
]
]̂
( ) ]̂
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 45
|
Vector Analysis: Gradient, Divergence and Curl ⃗⃗⃗
Q#11: Evaluate ̂
Solution: Let Now ⃗⃗⃗
[ ⃗⃗⃗ (
[ ⃗⃗⃗ (
⃗⃗⃗
)]
)]
̂ ̂
=
[ ⃗⃗⃗ (
= ⃗⃗⃗
[ (
= ⃗⃗⃗
[ { (
= ⃗⃗⃗
[ {(
)
= ⃗⃗⃗
[ {(
)
= ⃗⃗⃗
[(
=(
̂
() {
̂ )( ̂
) ̂
( ̂ ̂ ̂
̂
(
(
)
(
)
=(
)[
=(
) [{
=(
) [{
(
)
=(
)[
(
)
=(
)[
=(
)[
=(
)[
=(
)[
=(
)[
)
( (
)
̂ ̂
) [(
) ̂} ]
(
)
̂ )
}
)]
)̂ (
(
=
(
------(i)
)]
̂
)
B.Sc & BS Mathematics
( (
̂
(
)
̂
)
)
(
}
̂] )
̂
( ( (
̂]
)
)]
{
{
(
)
( }
̂} ]
) )
(
̂} ]
)
)
)
} }
(
{
)
(
{
(
)
)
) }]
]
] (
)] ]
] ]
= Hence
⃗⃗⃗
[ ⃗⃗⃗ (
)]
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 46
}]
Vector Analysis: Gradient, Divergence and Curl ̂
Q#12:If ( )
(i)Curl Let ⃗⃗
̂
(i)
̂
̂ (
⃗ )= 0
and ⃗⃗ is a constant vector then show that (iii) Curl (⃗⃗
̂ ̂
B.Sc & BS Mathematics
̂
(iv) ⃗⃗⃗
⃗ ) = 2⃗⃗
(
⃗
)
̂ ̂
Curl
Solution: Let ̂ ⃗⃗⃗
Curl
⃗
|= ̂ |
|
= ̂[
Curl
] ̂
|
̂[
]
= ̂[ =
̂ ̂
̂[
̂[
|
]
]
̂ ̂
=0 (
(ii)
⃗⃗ ) = 0
⃗ =
Solution:
̂
(
̂) ̂
̂
̂ Now
̂|
|
̂[
] ]
̂|
(
⃗⃗⃗
⃗)
(
=
[̂ |
=
{̂ [
=
{ ̂[
=
{
⃗)
̂|
̂[
]
|
̂|
| ̂[
]
̂
̂ |
|
̂
̂ ̂
|
]
̂ ̂
̂ (
|
)
|]
] ̂[
̂
̂[
]}
]}
̂}
= (
⃗ )=0
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 47
Vector Analysis: Gradient, Divergence and Curl Curl (⃗⃗⃗
(iii)
⃗⃗ ) = 2⃗⃗⃗ ̂
⃗
Solution:
⃗
̂ ̂
⃗ =|
⃗ = ̂[
B.Sc & BS Mathematics
]
|= ̂ |
̂[
̂|
|
̂[
]
]= ̂[
⃗ ) = ⃗⃗⃗
⃗ ) =⃗⃗⃗
(⃗⃗
= ̂|
| (
= ̂[ = ̂[
) ( ]
̂
=
)] – ̂ [
(
̂[
(
̂[
] ̂
(
)
|
(
̂
)]
)]
] ̂
(
̂) ̂
) ̂
⃗ =
̂ ̂ ̂
(
̂) ̂
̂
̂ ⃗⃗⃗
̂|
⃗ ) = 2⃗⃗
Solution : let
Now
]
|
|
)]
̂[
] ̂
̂
̂|
(
̂[ ̂
⃗⃗⃗
(iv)
̂[
̂[
|
(
)]
= ̂[
Curl (⃗⃗
⃗
|
]
̂ Now Curl (⃗⃗
̂|
|
(
⃗
⃗⃗⃗
)
(
⃗)
=
[̂ |
=
{̂ [
=
{̂[
̂ ̂
̂
|
|
|
]
̂ ̂
̂|
̂|
|
]
̂[
̂[
]
̂
|
(
|
)
|] ̂[
] ̂[
̂
]} =
]} {
̂
̂
̂}
= ⃗⃗⃗
(
⃗
) =0
Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 48
Vector Analysis: Gradient, Divergence and Curl Q#13: Show that ⃗⃗
⃗
Solution: : Let
̂
⃗ =
B.Sc & BS Mathematics , when ⃗⃗ = ⃗⃗⃗
is an Irrotational vector also find ̂ ̂ ̂
(
&
( )
(a>0)
=
̂) ̂
such that
̂
̂ ̂
For Irrotational vector, we have to prove Now Curl⃗⃗⃗⃗ ̂ Curl⃗⃗⃗⃗
⃗⃗⃗
⃗⃗⃗⃗
⃗
⃗⃗⃗
(
=
[̂ |
=
{̂ [
=
{̂[
⃗⃗⃗
)
(
⃗)
|
]
|
̂|
̂|
̂[
̂[
̂[
̂ ̂
|
(
|
)
|] ̂[
]
]
̂ |
|
]
̂ ̂
]} =
]} {
̂
̂} = ̂
Curl⃗⃗⃗⃗ = 0 ⃗
Hence prove that ⃗⃗ Now we have find ̂
is an Irrotational vector .
̂
̂= ̂
From (i) ,(ii) & (iii):
At
(
̂ ̂
Comparing coefficients of ̂ , ̂
=
⃗⃗ = ⃗⃗⃗
for this given condition is
⃗⃗⃗
Then ̂
̂ ̂
̂
∫
=
(
)
(
)-------(i)
∫
=
(
)
(
) -----(ii)
∫
=
(
)
=
(
)
( )=0
Hence equation (a) will become
⃗⃗
(
)-------(iii)
) =
( )=
=
-----(a)
=0 ( )= ( )
( )
( )
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 49
Vector Analysis: Gradient, Divergence and Curl Q#14: Find a, b, c so that ⃗
(
) ̂
B.Sc & BS Mathematics
(
)̂
) ̂ is Irrotational
(
vector. Solution: Given ⃗
(
) ̂
(
)̂
)̂
(
By using Given condition that ⃗ Curl⃗⃗
⃗⃗⃗
⃗⃗⃗⃗
̂
̂ ̂
|
| (
) (
̂| (
̂[
)
| )
(
(
) ) ( ̂[
)
̂| (
(
(
̂[
) (
)
)]
̂[
(
)
(
)
̂|
|
(
(
|= 0 )
) (
)]
̂[
(
)] )]
̂[
]
̂[
]
]
̂[
]
̂[
] ̂
̂ ̂
̂
̂ ̂
Comparing coefficients of ̂ ̂ & ̂ . c= 1 a= 4 b=2
Written & Composed by: Hameed Ullah, M.Sc Math (
[email protected]) GC Naushera
Page 50
Vector Analysis: Gradient, Divergence and Curl ( )
Q#15: Prove that . Solution: We know that ( )
Let
=
then ] ( )
[ =
[
( )]
=
[
( )
= [{
( )( )
=[
( ) {( ) ̂
]
( )
( )
[
] ( )
( )( )
( )
( )
( )
{
( )
̂ ̂
}
( ) ( )]
[
( )
[
( )
( )
( )]
[ ]
=[
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B.Sc & BS Mathematics
}
( )
( )( )
( ){
( ) }
then
( )
{
( ) ( )
}]
]
} ] ---------------(a) ------(i)
( ) Differentiate w. r. t x
2r
= 2x
Similarly
( )
Again differentiate w. r. t x
=
( )
=
=
Similarly
Putting values in Equation (a) ( )=
( ){
( )=
( ){
( )
=
( )( )
( )=
( )
( )
( ){ ( ){
{ }
}
( ){
}
( ){ }
( )=
( ){
}
(
} )
}
}
( ) Hence proved.
Written & Composed by: Hameed Ullah, M.Sc Math (
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