Chapter 8 :
The Integral Theorems of Vector Analysis
8 1
.
:
Green's Theorem
Simple and Elementary Regions and Their Boundaries :
Y
A (t G =
+
+ Bz" + G +
Bi -
broken down into simple components
-Be
A b5X
Green's intorem :
Lemma 1 :
/ Pax + Qay =
-(odx
D-y-simple region , -boundary of D
Proof :
D :
a = X= b , Q (x)
,
= y = P2(X)
"
holds for QO Ct =
C, + Be +
C+ B,
5 P 0 =
((X 3) axay =OP a , IX Y
/"(P(X , (x))-P(X , , (x)) = ax
( ,+ -
(X P , ,
(x))v(P(X Q (x))aX ( , ,
=
+
,
P(X y)aX
,
(
+ -
(X xx(x)) ,
+
/ P(X , 02(X))aX =
( ,
+
P(X y)ax
,
a
-P(x , Y
-)P(x Y)dx-
=
,
Cit
P(X Y)
, d
f P(X 4)ax
,
Green's Theorem
/ Pax +
Qay
/ -OP dx
: =
+
ex :
P(X y) ,
=
X ,
Q(X Y) ,
=
XY
D :
x3 + y = /
X =
COS(t) , y =
sin(t)
P(X Y) ,
=
cos(t) ,
dX =
-sin(t)dt
Q (x , y) =
cos(t) sin(t) , dy =
cos(t) It
Pdx +
Qdy =
/-cos(t)sint) + cost) sin(t
=
cost)
-
cost
= Q Y ,
/yayax- 0 by symmetry
Area :
/op Yay -yax
A= 1
Proof :
P(X y) -y ,
=
,
Q(X Y) ,
=
X
A
/- ) dx
=
, A
/#axdy
=
A =
/ ,
axdy =
A
923
213
eX :
y +
y43 =
X =
acos" (0) , y = asin It) ,
00 < In
A
/op Xay-yax
=
dy =
3 asin" (0) cos(0)
aX =
- 3acos" (0) sin(t)
A
=1ga sin (0) los"(A) U
+ 3 cos" 18) sin" (E) do
sin") cos(0) do
-
=
30 Sin (E) cos(8) doO
sin 12)
Sinf Lost
Sin10cos(f)
- =
2
In
: 302
J f O
sin (20) do
sin 120) :
1-cos(48)
2
34mI-cos(YE
:
San
·
O
,
do-san cost
=
-sin(4
Y
=
3/2n-0)-310-op
Y
=
302n
J
VectorF orm Using the Curl :
F =
Pi +
Q]
(a Pi + aj ·
as =
/) ,
curiF - dA =
(((XF) -G dA
(xF).
-
:
ex :
F(x Y) ,
=
(xy 2 , y + x)
(XF) ·=
- -y
:
X
lif X
2
1- ixy dy dx
xy2
X
↑ Xh
1 -
exydy =
y
-
= X -
x" x(x) xY)
- -
=
x
-
x2 x + x
-
↓ x -x x + XdX
X- +: 6-43
-
=
:
VectorF orm using Divergence :
: (9 b) ,
,
[(t) :
(x (t) Y(t) ,
n =
(y'(t) ,
-
X'(t) ,
F =
Pi + Q1
#xt()"
(y'(t))2 +
Finds If div =
da
The Integral Theorems of Vector Analysis
8 1
.
:
Green's Theorem
Simple and Elementary Regions and Their Boundaries :
Y
A (t G =
+
+ Bz" + G +
Bi -
broken down into simple components
-Be
A b5X
Green's intorem :
Lemma 1 :
/ Pax + Qay =
-(odx
D-y-simple region , -boundary of D
Proof :
D :
a = X= b , Q (x)
,
= y = P2(X)
"
holds for QO Ct =
C, + Be +
C+ B,
5 P 0 =
((X 3) axay =OP a , IX Y
/"(P(X , (x))-P(X , , (x)) = ax
( ,+ -
(X P , ,
(x))v(P(X Q (x))aX ( , ,
=
+
,
P(X y)aX
,
(
+ -
(X xx(x)) ,
+
/ P(X , 02(X))aX =
( ,
+
P(X y)ax
,
a
-P(x , Y
-)P(x Y)dx-
=
,
Cit
P(X Y)
, d
f P(X 4)ax
,
Green's Theorem
/ Pax +
Qay
/ -OP dx
: =
+
ex :
P(X y) ,
=
X ,
Q(X Y) ,
=
XY
D :
x3 + y = /
X =
COS(t) , y =
sin(t)
P(X Y) ,
=
cos(t) ,
dX =
-sin(t)dt
Q (x , y) =
cos(t) sin(t) , dy =
cos(t) It
Pdx +
Qdy =
/-cos(t)sint) + cost) sin(t
=
cost)
-
cost
= Q Y ,
/yayax- 0 by symmetry
Area :
/op Yay -yax
A= 1
Proof :
P(X y) -y ,
=
,
Q(X Y) ,
=
X
A
/- ) dx
=
, A
/#axdy
=
A =
/ ,
axdy =
A
923
213
eX :
y +
y43 =
X =
acos" (0) , y = asin It) ,
00 < In
A
/op Xay-yax
=
dy =
3 asin" (0) cos(0)
aX =
- 3acos" (0) sin(t)
A
=1ga sin (0) los"(A) U
+ 3 cos" 18) sin" (E) do
sin") cos(0) do
-
=
30 Sin (E) cos(8) doO
sin 12)
Sinf Lost
Sin10cos(f)
- =
2
In
: 302
J f O
sin (20) do
sin 120) :
1-cos(48)
2
34mI-cos(YE
:
San
·
O
,
do-san cost
=
-sin(4
Y
=
3/2n-0)-310-op
Y
=
302n
J
VectorF orm Using the Curl :
F =
Pi +
Q]
(a Pi + aj ·
as =
/) ,
curiF - dA =
(((XF) -G dA
(xF).
-
:
ex :
F(x Y) ,
=
(xy 2 , y + x)
(XF) ·=
- -y
:
X
lif X
2
1- ixy dy dx
xy2
X
↑ Xh
1 -
exydy =
y
-
= X -
x" x(x) xY)
- -
=
x
-
x2 x + x
-
↓ x -x x + XdX
X- +: 6-43
-
=
:
VectorF orm using Divergence :
: (9 b) ,
,
[(t) :
(x (t) Y(t) ,
n =
(y'(t) ,
-
X'(t) ,
F =
Pi + Q1
#xt()"
(y'(t))2 +
Finds If div =
da
