vector

Vectors 431
=
2
0
2 2
ˆ ˆ 2 ˆ 4 ˆ 3
4 2
ˆ
x y
dx yi xyj yk x yi x yj kˆ
2
2 4 3 2
= ˆ ˆ ˆ ˆ ˆ ˆ
(16i 8xj 16k 4xi 4x j 8 xk)
dx
0
2
5 3
2 4 4 8
= ˆ ˆ ˆ x
16 4 16 ˆ ˆ
x
xi x j xk i x j kˆ
5 3 0
128 64
= 32 ˆ 16 ˆ 32 ˆ ˆ 16 ˆ ˆ 32 iˆ
32kˆ
i j k i j k = = 32 (3iˆ
5 kˆ
)
5 3 5 3 15
EXERCISE 5.12
1. If F
= 2
(2x 3) z iˆ2xyj ˆ 4 xkˆ
, then evaluate FdV , where V is bounded by the plane
V
4
0
Ans.
x = 0, y = 0, z = 0 and 2x + 2y + z = 4. Ans. 8 3
2. Evaluate dV ,
V where = 45 x 2 y and V is the closed region bounded by the planes
4x + 2y + z = 8, x = 0, y = 0, z = 0 Ans. 128
3. If F = (2x 2
– 3z) iˆ
2xyj ˆ 4 xkˆ
, then evaluate FdV
, where V is the closed region bounded
V
8
by the planes x = 0, y = 0, z = 0 and 2x + 2y + z = 4.
Ans. ( ˆ ˆ)
3 j k
4. Evaluate (2 x
y) dV , where V is closed region bounded by the cylinder z = 4 – x 2 and the planes
V
x = 0, y = 0, y = 2 and z = 0. Ans. 80 3
5. If F 2
= 2 ˆ ˆ ˆ
xz i xj y k,
evaluate
F dV over the region bounded by the surfaces x = 0, y = 0,
y = 6 and z = x 2 , z = 4. Ans. (16iˆ3ˆj 48 kˆ
)
5.36 GREEN’S THEOREM (For a plane)
Statement. If (x, y), (x, y), and be continuous functions over a region R bounded
y
x
by simple closed curve C in x – y plane, then
( dx
dy)
=
dx dy
C
R
x
y
(AMIETE, June 2010, U.P., I Semester, Dec. 2007)
Proof. Let the curve C be divided into two curves C 1
(ABC) and C 1
(CDA).
Let the equation of the curve C 1
(ABC) be y = y 1
(x) and equation of the curve C 2
(CDA) be
y = y 2
(x).
Let us see the value of
dx dy = xc
y
y
2
( x)
dy dx
R y
xa y y ( x)
1 y
= c 2 ( )
( , )
y
x y
y x
dx
a
y y ( x)
1
c
a
c
= ( x, y2) ( x, y1)
dx = ( x, y2) dx ( x, y1)
dx
a
a
c
=
( x, y2) dx ( x, y1)
dx
c
a
=
( x , y ) dx ( x , y ) dx
c2 c1
= – ( , )
c
c
a
x y dx