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Chapter 16 – Vector Calculus16.4 Green’s Theorem

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 2

Introduction 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.

George Green(1793 – 1841)

16.4 Green’s Theorem 3

IntroductionGreen’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 4

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 5

Green’s TheoremLet 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

CD

Q PPdx Qdy dA

x y

16.4 Green’s Theorem 6

Green’s Theorem NotesThe notation

is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve C.

orC CPdx Qdy Pdx Qdy

16.4 Green’s Theorem 7

Green’s Theorem NotesAnother notation for the positively oriented

boundary curve of D is ∂D.

So, the equation in Green’s Theorem can be written as equation 1:

DD

Q PdA Pdx Qdy

x y

16.4 Green’s Theorem 8

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:

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.

'( ) ( ) ( )b

aF x dx F b F a

16.4 Green’s Theorem 9

AreaGreen’s Theorem gives the

following formulas for the area of D: 1

2C C C

A xdy ydx xdy ydx

16.4 Green’s Theorem 10

Example 1 – pg. 1060 #6Use Green’s Theorem to evaluate the line

integral along the given positively oriented curve.

2cos sin

is the rectangle with verticies

0,0 , 5,0 , 5,2 , and 0,2

C

ydx x ydy

C

16.4 Green’s Theorem 11

Example 2 – pg. 1060 #8Use Green’s Theorem to evaluate the line

integral along the given positively oriented curve.

2 4 2 2

2 2 2 2

2

is the boundary of the region between the circles

1 and 4

x

C

xe dx x x y dy

C

x y x y

16.4 Green’s Theorem 12

Example 3 – pg. 1060 #12Use Green’s Theorem to evaluate . Check the orientation of the curve before

applying the theorem.

2 2( , ) cos , 2 sin

is the triangle from

0,0 to 2,6 to 2,0 to 0,0

x y y x x y x

C

F

CdF r