Green’s Theorem

Assumptions:
J. Lewis

F ( x, y )  P ( x, y ) i  Q ( x, y ) j
derivatives on an open, simply connected set
C

P and Q
have continuous partial
D.
is a piecewise smooth, simple, closed curve in
D. C
encloses a region
D which is simply
connected.
Since Green’s Theorem I involves the curl of the vector field, we need to know what the curl is.
Curl of a 2-d Vector Field

If
F is the velocity vector of a fluid, then these vectors are tangent to the flow lines of the fluid.
We wish to describe the tendency of the vector field to swirl around a pole through a point,
( x0 , y0 ).
The pole is orthogonal to the plane of the swirling, the x,y- plane, so the pole is parallel to the z-axis.


This means the direction of our curl vector is k or  k . (In 3-D, which we do later, we need to specify
the direction of the pole). If the swirling is counterclockwise, we want the curl to be a positive number

times
k

. If the swirling is clockwise, it should be a negative number times
Since we want to describe the swirling, we will look at some examples.

1.


F ( x, y )  x i  y j


about (0,0).

2.
F ( x, y )   y i  x j
3.
F ( x, y )  (

x y
2

) i(
about (0,0).
x y
2

) j
about (0,0).
k
.


 F  dr
Consider
for each of examples 1 and 2, for
Ca : x 2  y 2  a 2 .
Ca
For example 1 the integral is 0.
For example 2 the integral is
2 a
2


1
F  dr  2 .
2 
a C
=twice the area of the enclosed disk.
a
In general we get a function of
a, x0 , y0 on the right. We define the curl of the vector field as
 


1
curl F ( x 0 , y 0 )   lim
 F  dr  k
a0  a 2 C


C a : ( x  x0 ) 2  ( y  y 0 ) 2  a 2 .

for
a

Can we find a simpler expression for
curl F
and see that this limit even exists?

Consider the linear case:




F ( x , y )  ( p1 x  p 2 y ) i  ( q1 x  q 2 y ) j



 F  dr   F1  dr   F2  dr
Ca
Ca
F1
is conservative and

where




2

. We have

2
 F  dr  ( q1  p2 ) a
Ca
 



1
curl F ( x 0 , y 0 )   lim
 F  dr  k  ( q1  p 2 ) k
a0  a 2 C



a
In general, the vector field is not necessarily linear but it can be shown that
 Q P  
curl F ( x 0 , y 0 )  

k

x

y




F1  p1 x i  q 2 y j , F2  p2 y i  q1 x j .
 F2  dr  ( q1  p 2 ) a
Ca
So

.
Ca



Green’s Theorem:
 Q P 

dA , where
 F  dr   
y 
C
D  x

Green’s Theorem I
Under the assumptions on page 1,

C is traversed once counterclockwise.
Green’s Theorem I is a statement about the integral of the tangential component of the vector field
around
C.

Verify This statement in the case:


F ( x , y )  2 xy i  3 x y j
2
2
and
C
is the boundary of the
triangle with vertices (0,0), (1,1) and (0,1) traversed once counterclockwise.
Green’s Theorem II The 2nd statement of Green’s theorem is a statement about the integral of the
normal component of the vector field around
C.

Let
Let
n
be the outward unit normal to



C , keeping D on the left as we go counterclockwise along C .

F  Q i  P j .
 



F  n  F T
Applying Green’s Theorem I to


 






F
we have Green’s Theorem II,
P Q

dA
y
D x
 F  n ds   F  T ds   F  dr  
C
P Q

x y
C
C

is called the divergence of

F , written div F , and measures the tendency to flow
outward from the point.
Verify the 2nd statement of Green’s Theorem for the examples on page 1.