UNIT 2 Vector Calculus Class notes - Green’s, Stoke’s and Gauss Divergence theorem (Part-3)
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Module
18MAB102T
Institution
SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
Unit II -Vector Calculus
Class notes for Course 18MAB102T - Advanced Calculus and Complex Analysis, Unit 2.
Our notes deliver a thorough exploration of vector calculus principles, including gradient, divergence, and curl, as well as the crucial theorems such as Green's, Stokes', and the Divergen...
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
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
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
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
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