This document discusses vector calculus theorems including Stokes' theorem and the divergence theorem. Stokes' theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the vector field over any surface bounded by the curve. The divergence theorem relates the volume integral of the divergence of a vector field over a volume to the surface integral of the vector field over the boundary surface of that volume. The document provides proofs of these theorems and examples of their applications to problems involving conservative vector fields and evaluating integrals.

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topics: Vector Integration
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2. Line Integration
P
Q
Closed, non-
intersecting line
forms a single loop
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3. Surface Integration
Surface
Closed, non-intersecting line/loop can form an open surface
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4. Volume Integration
Closed, non-intersecting
surface forms volume
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5. Line Integration Surface Integration
If 𝑨 is continuous and differentiable at any open surface S formed
by a closed, non-intersecting curve C
( ) ˆ
. .
S
A dr A ndS
= 
 
( )
. .
S
A dr A dS
= 
 
This is Stoke’s theorem
where ෝ
𝒏 is the positive outward normal on the surface S
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6. Statement of Stoke’s theorem
( )
. .
S
A dr A dS
= 
 
Closed line integration of any continuous and differentiable vector is
numerically equal to the surface integration of curl of that vector where
surface is formed by that non-intersecting closed curve
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7. Proof of Stoke’s theorem
X
Y
Z
A (x, y) B (x+Δx, y)
C (x+Δx, y+Δy)
D (x, y+Δy)
Open surface formed by the closed path
Closed path ABCDA
𝑷 is continuous and differentiable
at the surface
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8. Proof of Stoke’s theorem
Value of the vector from A to B  
x
P x
= 
Value of the vector from B to C
y
y
P
P x y
x

 
 
= +  
 
 

 
 
y
y
P
P y x y
x

 
 
 
=  +  
 
 
  
 
 
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9. Proof of Stoke’s theorem
Value of the vector from C to D
  x
x
P
P x y x
y
 
 

= −  −  
 
 

 
 
x
x
P
P y x
y
 
 

= − +  
 
 

 
 
Value of the vector from C to D
Value of the vector from D to A
y
P y
 
= − 
 
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10. Proof of Stoke’s theorem
Value of the vector over ABCDA
y x
P P
x y
x y

 
   

= −  
 
   
 
 
 
 
( ) .
z
P dS
= 
( ) y x
z
P P
P
x y

 
   

 = −
 
   
 
 
 
 
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11. Proof of Stoke’s theorem
X
Y
O
For plane XOY
( )
. .
XOY z
S
P dr P dS
= 
 
For plane YOZ
( )
. .
YOZ x
S
P dr P dS
= 
 
For plane ZOX
( )
. .
ZOX y
S
P dr P dS
= 
 
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12. Proof of Stoke’s theorem
Combining all the planes
( )
. .
S
P dr P dS
= 
 
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13. Stoke’s theorem
for every closed curve is that 𝑨 is irrotational
. 0
Adr =

Prob 1: Show that necessary and sufficient condition that
Soln
𝑨 is irrotational ( ) 0
A
 =
( )
. . 0
S
A dr A dS
=  =
 
So the condition is sufficient
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14. Stoke’s theorem
Let
( ) ˆ
A n

 = β is positive constant
( )
ˆ ˆ
. . 0
S
Adr n n dS

 = 
 
This contradicts the hypothesis
( ) 0
A
 = is the necessary condition also
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15. Conservative Vector Field
If 𝑷 is irrotational, then the vector field is conservative
0
P
 =
If
Then 𝑷 is conservative vector field
P 
 = 
We know that
0

 =
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16. Conservative Vector Field
Prob 2: Show that for conservative field, work done is path independent
Soln
Work done
2
1
.
P
P
W F dr
= 
2
1
.
P
P
W dr

= 

( )
2
1
ˆ ˆ
ˆ ˆ ˆ ˆ
. ( )
P
P
W i j k dxi dyj dzk
x y z
  
 
  
= + + + +
 
  
 

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17. Conservative Vector Field
2
1
P
P
W dx dy dz
x y z
  
 
  
= + +
 
  
 

W d
=
2 1
P P
W  
= −
Therefore, work done depends on the potentials of starting and end points,
but not on the path joining them
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18. If 𝑨 is continuous and differentiable at any volume V formed by a
closed, non-intersecting surface S
( ) ˆ
. .
V
S
A dV A ndS
 =
 
( )
. .
V
S
A dV A dS
 =
 
where ෝ
𝒏 is the positive outward normal on the surface S
This is Divergence theorem
Volume Integration Surface Integration
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19. Statement of Divergence theorem
( )
. .
V
S
A dV A dS
 =
 
Volume integration of divergence of any continuous and differentiable
vector over any closed surface is numerically equal to the surface
integration of that vector where volume is formed by the non-intersecting
closed surface
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20. Proof of Divergence theorem
Z
X
Y
O
A (x, y) B (x+Δx, y)
C (x+Δx,
y+Δy)
Net outward flux while moving From A to B
1
2
x
x
P
P dx dydz
x

 
 
+  
 

 
 
Net outward flux while moving From B to C
1
2
y
y
P
P dy dzdx
y

 
 
+
 
 

 
 
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21. Proof of Divergence theorem
A (x, y) B (x+Δx, y)
C (x+Δx,
y+Δy)
O
Z
X
Y
Net outward flux while moving From C to D
1
2
x
x
P
P dx dydz
x

 
 
−  
 

 
 
Net outward flux while moving From D to A
1
2
y
y
P
P dy dzdx
y

 
 
−
 
 

 
 
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22. Proof of Divergence theorem
Net outward flux along X-direction
1
2.
2
x x
P P
dxdydz dxdydz
x x
 
   
=
   
 
   
Net outward flux along Y-direction
1
2.
2
y y
P P
dydzdx dydzdx
y y
 
   
=
   
 
   
Net outward flux along Z-direction
1
2.
2
z z
P P
dzdxdy dzdxdy
z z
 
   
=
   
 
   
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23. Proof of Divergence theorem
Total outward flux
. y
x z
S
P
P P
P dS dxdydz
x y z

 
 
= + +
 
  
 
 
( )
. .
V
S
P dS P dV
= 
 
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24. Prob 3: Evaluate
Divergence theorem
( )
. .
V
S
r dS r dV
= 
 
.
S
r dS

Soln
Using Divergence theorem
ˆ ˆ
ˆ ˆ ˆ ˆ
. .( )
V
S
r dS i j k xi yj zk dV
x y z
 
  
= + + + +
 
  
 
 
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25. Divergence theorem
.
V
S
r dS x y z dV
x y z
 
  
= + +
 
  
 
 
. 3
V
S
r dS dV
=
 
. 3
S
r dS V
=

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26. Divergence theorem
Prob 4: Show that
( ) ˆ.
V
S
dV n dS
 
 =
 
Soln
Using Divergence theorem
( )
. .
V
S
A dV A dS
 =
 
Let
A B

= 𝑩 is constant
Let
A B

=
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27. Divergence theorem
( )
. .
V
S
B dV B dS
 
 =
 
( ) ˆ
. .
V
S
B dV B ndS
 
 =
 
( ) ˆ
. ( . )
V
S
B dV B n dS
 
 =
 
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28. Divergence theorem
( ) ˆ
. . ( )
V
S
B dV B n dS
 
 =
 
( ) ˆ
( )
V
S
dV n dS
 
 =
 
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29. Divergence theorem
Prob 5: Show that ( ) ˆ
( )
V
S
A dV n A dS
 = 
 
Soln
Using Divergence theorem ( )
. .
V
S
P dV P dS
 =
 
Let
P A B
= 
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30. Divergence theorem
( )
.[ ] [ ].
V
S
A B dV A B dS
  = 
 
( ) ˆ
. . [ ]
V
S
B A dV B n A dS
 = 
 
( ) ˆ
[ ]
V
S
A dV n A dS
 = 
 
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31. Green’s theorem
Let
( )
. .
V
S
P dV P dS
 =
 
From Divergence theorem
P  
= 
( )
. .
V
S
dV dS
   
  = 
 
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32. Green’s theorem
( )
2
. .
V
S
dV dS
     
 +   = 
 
………. (2)
This is Green’s First Identity
Interchanging φ and ψ
( )
2
. .
V
S
dV dS
     
 +   = 
 
………. (1)
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33. Green’s theorem
This is Green’s Second Identity
Substituting (2) from (1)
( ) ( )
2 2
.
V
S
dV dS
       
 −  =  − 
 
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34. Helmhotz’s theorem
A vector field can be uniquely defined by its divergence and curl
Proof
We consider two vectors 𝑷 and 𝑸 defined, continuous and differentiable
In a closed, non-intersecting surface S which forms the volume V
ˆ ˆ
. .
P n Q n
=
ˆ
( ). 0
P Q n
− = ………. (1)
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35. Helmhotz’s theorem
At any arbitrary point
. .
P Q
 = 
.( ) 0
P Q
 − = ………. (2)
Again, at that point
P Q
 = 
( ) 0
P Q
 − = ………. (3)
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36. Helmhotz’s theorem
Let
( )
R P Q
= −
Since 𝑹 is both solenoidal and irrotational
R 
= 
where φ is arbitrary scalar function
………. (4)
. 0

  =
From (2)
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37. Helmhotz’s theorem
2
0

 =
Using Green’s First Identity
( )
2
. .
V
S
dV dS
     
 +   = 
 
For φ = ψ
( )
2
. .
V
S
dV dS
     
 +   = 
 
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38. Helmhotz’s theorem
( )
2
.
V
S
dV dS
  
 = 
 
( )
2
.
V
S
dV R dS
 
 =
 
ˆ
( ). 0
P Q n
− =
From (1)
ˆ
. 0
R ndS =
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39. Helmhotz’s theorem
ˆ
. 0
R ndS =

( )
2
0
V
dV

  =

0

 =
0
P Q
− =
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40. Helmhotz’s theorem
P Q
=
Therefore, the vector is unique which can be defined
by its divergence and curl
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