WTW258: LU 3.4: GREEN’S THEOREM Lecture notes

3. 4 .
Green 's theorem
that doesn't intersect
A simple c closed ) curve
-




itself ×
↳ +
d
✓
intersects

•


A region is a simple connected region
,
if the

boundary consists of one simple closed curve :


✓
+ ×




•
ppoossiittiiwee orientation of a simple , closed curve

anti-clockwise direction
①
.




btw line
☒ rreeeenn $ Tthheeooroemm relation integrate
" -
.




double integral

ftp.i.EE-atiisitiitti: :
2 3

positive
.
.




let c be a orientated piecewise smooth
close# curve in XY plane that forms the
-




boundary of a
Region R :

@


④

, Then :


{ Pdx Ady If an Pydn
-
. _
-




R
or alternatively
{ F. di
JI Qu
Py DA f- P a >
with
- - = -
,




R


③ different ways to determine

{ F- di with Cin R2 :
-




c ( Lu 3. 2)
1. Parametrize the curve
C : FC 1- 5- < ✗ It)
, yet ) >
,
t E ca b
, ]

{ E. OF
-
-




Jab Écrct) ) F -
'
Lt) dt


theorem of line integrals if
2. Fund .




F- =
conservative v. f. that is F- =D f-
then
{ E. di Tf f- ( Fcb) ) f- ( F (d) )
= = -




Green 's theorem if [ is closed csimp e)
3.
/

curve , orientated counter clock -
.




Spd ≥ ' Qdy F- di
{ J) an byd A ICU 3. 4)
- -
-
-




R