This document discusses Green's theorem, which relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It presents the statement of Green's theorem, which equates the line integral of P dx + Q dy around C to the double integral of (∂Q/∂x - ∂P/∂y) over D. An example problem demonstrates using Green's theorem to evaluate a line integral by transforming it into a double integral. Verifying the equality of the two approaches confirms Green's theorem for the given region.

1. GREEN THEOREM
(TRANSFORMATION OF DOUBLE INTEGRALS TO LINE INTEGRALS)
COMPLEX VARIABLE AND LAPLACE TRANSFORM
PRESENTED BY: Sarwan Ahmed Ursani (18CH101)
ASSIGNED BY: Sir Ayaz Siyal

2. INTRODUCTION
 This law was proposed by George Green in 1828 A.D. and is named after him.
 In mathematics, Green's theorem gives the relationship between a line integral around,
a simple closed curve C and a double integral over the plane region D bounded by C.
 The Green theorem is used to transform double integrals over a plane region into line integral
over the boundary of the region.
 Identities derived from Green's theorem play a key role in reciprocity in electromagnetism
 It makes the contour integral easier.
NOTE: In the mathematical field of complex analysis, contour integration is a method of
evaluating certain integrals along paths in the complex plane.

3. STATEMENT OF GREEN THEOREM
 Let R be a closed bounded region in the xy- plane, whose boundary C consists of finitely many
smooth curves.
 Let P(x,y) and Q(x,y) be the continuous function having continuous partial derivative.
In region R and on its boundary C, Green theorem states that
(𝑃 𝑑𝑥 + 𝑄 𝑑𝑦) = (
𝜕𝑄
𝜕𝑥
−
𝜕𝑃
𝜕𝑦
)dy dx
𝜕𝑄
𝜕𝑥
+
𝜕𝑃
𝜕𝑦

4. PROOF OF GREEN THEOREM
 Consider a simple closed curve C bounding the region R as show in figure. Let
us divide the curve C into two parts ACB and ADB and let us suppose that the
equations of these curves be y= f1(x) and y = f2(x) respectively.
Now R is the region bounded by C, hence
𝜕𝑃
𝜕𝑦
dy dx = =
= 𝑎
𝑏
[𝑃 ( 𝑥, 𝑓1 𝑥 ) − 𝑃 (𝑥, 𝑓2 𝑥 )]𝑑𝑥
= − 𝑎
𝑏
𝑃 𝑥, 𝑓1 𝑥 𝑑𝑥 − 𝑏
𝑎
𝑃 (𝑥, 𝑓2 𝑥 )𝑑𝑥

5. = - 𝑃 𝑑𝑥
Hence
𝜕𝑃
𝜕𝑦
dy dx =− 𝑃 𝑑𝑥 … … … (𝑖)
Now consider the curve CAD and CBD. Let their equations be y = g1(x) and y = g2(x) this implies
that x = g1
-1(y) = G1(y) and x = g2
-2 (y) = G2(y)
𝜕𝑄
𝜕𝑥
dy dx = 𝑑𝑦 = 𝑑𝑦
= 𝐶
𝑑
[𝑄 (𝐺2 𝑦 , 𝑦) − 𝑄 (𝐺1 𝑦 , 𝑦)]𝑑𝑦
=
𝐶
𝑑
𝑄 𝐺2 𝑦 , 𝑦 𝑑𝑦 −
𝑑
𝑐
𝑄 𝐺1 𝑦 , 𝑦 𝑑𝑦

6. =
𝑑
𝑐
𝑄 𝐺1 𝑦 , 𝑦 𝑑𝑦 +
𝑐
𝑑
𝑄 𝐺2 𝑦 , 𝑦 𝑑𝑦
= - 𝑄 𝑑𝑦
Hence
𝜕𝑄
𝜕𝑥
dy dx = 𝑄 𝑑𝑦 … … … … … (𝑖𝑖)
Subtracting Eq (i) From Eq (ii)
𝑃 𝑑𝑥 + 𝑄 𝑑𝑦 =
𝜕𝑄
𝜕𝑥
dy dx −
𝜕𝑃
𝜕𝑦
dy dx
OR
𝑃 𝑑𝑥 +𝑄 𝑑𝑦 = (
𝜕𝑄
𝜕𝑥
-
𝜕𝑃
𝜕𝑦
) 𝑑𝑦𝑑𝑥
This Proves Green Theorem

7. Exercise Problem
Q: Verify the Green’s theorem for 𝟐𝒙𝒚 − 𝒙𝟐 𝒅𝒙 + 𝒙 + 𝒚𝟐 𝒅𝒚 where C is simple closed curve enclosed by
the parabolas y = x2 and x = y2 .
First we will solve this problem by contour integration.
We have two equation,
y = x2 …(i) and x = y2 …(ii)
x4 = x
x4 – x = 0
x(x3-1) = 0
We get
x = 0, x = 1
Now when x =0 , y = 0 & x= 1, y = 1.
We can say that the point of intersection between the two given curve are (0,0) and
(1,1).

8. Along C1
y = x2 so that dy = 2x and x varies from 0 to 1
=
0
1
[ 2𝑥. 𝑥2 − 𝑥2 𝑑𝑥 + (𝑥 + 𝑥4)(2𝑥𝑑𝑥)]
=
0
1
[ 2𝑥3 − 𝑥2 + (2𝑥2 + 2𝑥5) ]𝑑𝑥
=
0
1
(2𝑥5 + 2𝑥3 + 𝑥2 ]𝑑𝑥
=
1
3
+
1
2
+
1
3
=
2 + 3 + 2
6
=
7
6

9. Along C2
x = y2 so that dx = 2y and y varies from 1 to 0
=
1
0
[ 2𝑦. 𝑦2 − 𝑦4 2𝑦𝑑𝑦 + (𝑦2 + 𝑦2)(𝑑𝑦)]
=
1
0
[ 4𝑦4 − 2𝑦5 + 2𝑦2 𝑑𝑦
= [ 0 − (
4
5
−
1
3
+
2
3
) =
5 − 12 − 10
15
= −
17
15
Now, C = C1 + C2
=
7
6
−
17
15
=
35 − 34
30
=
1
30

10. Now, using Green Theorem
We can solve this problem easily
Here P = 2𝑥𝑦 − 𝑥2 & Q = 𝑥 + 𝑦2
A/C to Green Theorem
𝑃 𝑑𝑥 +𝑄 𝑑𝑦 = (
𝜕𝑄
𝜕𝑥
-
𝜕𝑃
𝜕𝑦
) 𝑑𝑦𝑑𝑥
𝜕𝑄
𝜕𝑥
= 𝟏 &
𝜕𝑃
𝜕𝑦
= 2𝑥
0
1
(√𝑥 − 𝑥2)(1−2x) dx

11. =
2
3
−
4
5
−
1
3
+
1
2
=
20 − 24 − 10 + 15
30
=
1
30
This is proves the green theorem for the given contour integral over the region R.