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 computed as the double integral over region D, with the path integral traversed counterclockwise. The theorem is proved by dividing the bounding curve C into two parts and expressing the line integrals around each part in terms of their bounding functions, showing they are equal to the double integral over region D.

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

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) and y = f2(x) respectively.
Now R is the region bounded by C, hence