Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It states that a line integral around the boundary of a plane region D can be calculated as the double integral over region D. The theorem allows contour integrals to be transformed into double integrals over the interior region, and plays a key role in reciprocity in electromagnetism. The document provides the statement of Green's theorem and gives a proof of the theorem by dividing the bounding curve C into two parts and considering the equations of the curves.

1. GREEN’S THEOREM
By
Ram Nivas Sonkar
M.Sc. First Semester(2022-23)
Enroll. No. 227/22
Mentor
(Ass. Prof.) Dr. Devendra Singh
School of Physical and Decision Science
Department of Physics
Baba saheb Bhimrao Ambedkar Univesity
( A Central University)
Lucknow-226025.UP

2. Content :
 Introduction
 Statement of Green’s Theorem
 Proof of Green’s Theorem
Exercise Problem
 Application of Green’s Theorem

3. Introduction
 This Law was proposed by George Green in 1828 .
 In mathematics, Green’s theorem gives the relationship
between a line integral around a simple 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 over the boundary of the region.
 Identities derived from Green’s theorem play a key role in
reciprocity in electromagnetism.
 Its makes the contour integral easier.

4. STATEMENT OF GREEN’S THEOREM
Green’s Theorem States that “A Line Integral around the boundary
of the Plane region D can be computed as the Double integral over
the region D”. Where the Path integral is traversed anti- clockwise.
 Let R be a closed bounded region in the x-y plane
 whose boundary C consist of finitely many
smooth curves.
 Let P(x,y) and Q(x,y) be the continuous function
having continuous partial derivative.
y
P
x
Q





In region R and on its boundary C, Green’s theorem states that:
dydx
y
P
x
Q
Qdy
Pdx )
(
)
(
  







5. Proof of Green’s Theorem
Consider a simple closed curve C bounding the region R as shown in fig. Let us
divide the curve C into two part ACB and ADB and Let us suppose that the
equations of these curve be y=f1(x) .
In region R and on its boundary C, Green’s theorem states that
dydx
y
P
x
Q
Qdy
Pdx )
(
)
(
  





