Chapter 16 – Vector Calculus
16.4 Green’s Theorem
Objectives:
 Understand Green’s
Theorem for various regions
 Apply Green’s Theorem in
evaluating a line integral
16.4 Green’s Theorem
1
Introduction
George Green
(1793 – 1841)

George Green worked in his father’s bakery
from the age of 9 and taught himself
mathematics from library books.

In 1828 published privately what we know
today as Green’s Theorem. It was not widely
known at the time.

At 40, Green went to Cambridge and died 4
years after graduating.

In 1846, Green’s Theorem was published
again.

Green was the first person to try to formulate
a mathematical theory of electricity and
magnetism.
16.4 Green’s Theorem
2
Introduction

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.
◦ We assume that D
consists of all points
inside C as well as
all points on C.
16.4 Green’s Theorem
3
Introduction

In stating Green’s Theorem, we use the convention:
◦ The positive orientation of a simple closed curve C
refers to a single counterclockwise traversal of C.
◦ Thus, if C is given
by the vector
function r(t),
a ≤ t ≤ b, then the
region D is always
on the left as the
point r(t) traverses
C.
16.4 Green’s Theorem
4
Green’s Theorem

Let C be a positively oriented, piecewise-smooth,
simple closed curve in the plane and let D be the
region bounded by C.
◦ If P and Q have continuous partial derivatives
on an open region that contains D, then
 Q P 
P
dx

Q
dy


dA


C
D  x y 
16.4 Green’s Theorem
5
Green’s Theorem Notes

The notation

C
P dx  Q dy or

C
P dx  Q dy
is sometimes used to indicate that the line
integral is calculated using the positive
orientation of the closed curve C.
16.4 Green’s Theorem
6
Green’s Theorem Notes

Another notation for the positively oriented
boundary curve of D is ∂D.

So, the equation in Green’s Theorem can
be written as equation 1:
 Q P 

dA

P
dx

Q
dy


D  x y  D
16.4 Green’s Theorem
7
Green’s Theorem and FTC

Green’s Theorem should be regarded as the counterpart of
the Fundamental Theorem of Calculus (FTC) for double
integrals.

Compare Equation 1 with the statement of the FTC Part 2
(FTC2), in this equation:

b
a

F '( x) dx  F (b)  F (a)
In both cases,
◦ There is an integral involving derivatives (F’, ∂Q/∂x, and ∂P/∂y)
on the left side.
◦ The right side involves the values of the original functions
(F, Q, and P) only on the boundary of the domain.
16.4 Green’s Theorem
8
Area

Green’s Theorem gives the following
formulas for the area of D:
1
A   xdy    ydx   xdy  ydx
2C
C
C
16.4 Green’s Theorem
9
Example 1 – pg. 1060 #6

Use Green’s Theorem to evaluate the line integral
along the given positively oriented curve.
 cos ydx  x
2
sin ydy
C
C is the rectangle with verticies
 0, 0  ,  5, 0  ,  5, 2  , and  0, 2 
16.4 Green’s Theorem
10
Example 2 – pg. 1060 #8

Use Green’s Theorem to evaluate the line integral
along the given positively oriented curve.
 xe
2 x
dx   x  2 x y  dy
4
2
2
C
C is the boundary of the region between the circles
x  y  1 and x  y  4
2
2
2
2
16.4 Green’s Theorem
11
Example 3 – pg. 1060 #12

Use Green’s Theorem to evaluate F dr .
C
 Check the orientation of the curve before
applying the theorem.

F( x, y )  y cos x, x  2 y sin x
2
2
C is the triangle from
 0, 0  to  2, 6  to  2, 0  to  0, 0 
16.4 Green’s Theorem
12