UNIT 2 Vector Calculus Class notes - Green’s, Stoke’s and Gauss Divergence theorem (Part-3)

18MAB102T- ADVANCED
CALCULUS AND COMPLEX
ANALYSIS; Unit II (Part-3) -
Green’s, Stoke’s and Gauss
Divergence theorem

Dr. Sahadeb Kuila
Assistant Professor
Department of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Outline




1 Green’s theorem


2 Stoke’s theorem


3 Gauss divergence theorem




2/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Statement (Green’s theorem):


Let C be a positively oriented, piecewise smooth, simple,
closed curve and let R be the region enclosed by the curve C
in the xy -plane. If P(x , y ) and Q(x , y ) have continuous first
order partial derivatives on R, then
!
I ZZ
∂Q ∂P
Pdx + Qdy = − dxdy .
C ∂x ∂y
R




3/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Applications of Green’s theorem
Example 1:
I
Use Green’s theorem to evaluate xydx + x 2 y 3 dy , where C
C
is the triangle with vertices (0, 0), (1, 0), (1, 2) with positive
orientation.
Solution: Let P = xy , Q = x 2 y 3 and the positive orientation
curve C is as shown in the figure.




4/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Applications of Green’s theorem
Using Green’s theorem,

I I
xydx + x 2 y 3 dy = Pdx + Qdy
C C
!
ZZ
∂Q ∂P ZZ
= − dxdy = (2xy 3 − x )dxdy
∂x ∂y
R R
Z 1 Z 2x Z 1" 4 #2x
3 xy
= (2xy − x )dydx = − xy dx
0 0 0 2 0
#1
4x 6 2x 3
Z 1 "
5 2 2
= (8x − 2x )dx = − = .
0 3 3 0
3


5/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Applications of Green’s theorem
Example 2:
Verify
I Green’s theorem in the plane for
[(xy + y 2 )dx + x 2 dy ], where C is the closed curve of the
C
region bounded by y = x and y = x 2 .
Solution: Let P = xy + y 2 , Q = x 2 and the positive
orientation curve C is as shown in the figure. The curves
y = x and y = x 2 intersect at (0, 0) and (1, 1).




6/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur

, Green’s theorem
Stoke’s theorem
Gauss divergence theorem


Applications of Green’s theorem
Using Green’s theorem,
I I
[(xy + y 2 )dx + x 2 dy ] = Pdx + Qdy
C C
!
ZZ
∂Q ∂P ZZ
= − dxdy = (2x − x − 2y )dxdy
∂x ∂y
R R
ZZ Z 1Z x
= (x − 2y )dxdy = (x − 2y )dydx
0 y =x 2
R
Z 1h ix Z 1
= xy − y 2 2
dx = (x 4 − x 3 )dx
0 y =x 0
#1
x5 x4
"
1
= − =− .
5 4 0
20
7/ 20 Dr. Sahadeb Kuila Dept. of Mathematics, SRMIST, Kattankulathur