vector

Vectors 463
a
2 2 2
bc x b cx bc x
= 2 = [a
2 2 2
2 bc + ab 2 c + abc 2 ]
0
= abc (a + b + c) ...(A)
To evaluate Fnds ˆ , where S consists of six plane surfaces.
S
= ˆ ˆ ˆ
OABC
DEFG OAFG
Fnds ˆ
S
Fnds
OABC
ˆ =
F nds ˆ
DEFG
Fnds Fnds Fnds
Fnds ˆ Fnds ˆ Fnds ˆ
=
=
=
=
=
BCDE ABEF OCDG
OABC
ab
00
ab
00
2 2 2
{( x yzi ) ˆ ( y xz) ˆj ( z xy) kˆ}( kˆ)
dxdy
2
( z xy)
dxdy
(0 xy)
dx dy =
DEFG
ab
ab
2
( z xy)
dxdy =
00
00
a
2
b a
2 xy
c y dx
2
=
0 0
0
2 2
2 x b
= cbx
4
Fnds
OAFG
ˆ =
=
=
a
OAFG
c
0 0
ˆ
2
a
0
=
OAFG
( y zx)
dxdz
dx
(0 zx)
dz =
2 2
a b
4
...(1)
2 2 2
{( x yz) iˆ
( y xz) ˆj ( z xy) kˆ}( kˆ)
dxdy
2
( c xy)
dxdy
2 xb
cb
2
2 2
2
dx
2 a b
abc ...(2)
4
{( x yz) iˆ ( y zx) ˆj ( z xy) k}( ˆj)
dxdz
2 2 2 ˆ
a
2
xz
dx
2
=
c
0 0
a xc
2
dx =
2
0
x c
4
F nds 2 2 2
= {( x yz ) i ˆ ( y zx ) ˆ j ( z xy ) k ˆ}
ˆ jdxdz
BCDE =
Fnds
ABEF
=
a c
a
2
c
2
2 xz
dx ( b xz)
dz =
b z
2
0 0
0 0
2 2
a
2 2
2 x c 2 ac
2 2
dx =
a
0
a
0
=
2 2
a c
4
BCDE
2 xc
b c
2
2
...(3)
( y xz)
dxdz
2
dx
= b cx = ab c ...(4)
4
4
0
2 2 2
ˆ
ˆ
= {( ) ˆ ( ) ˆ ( ) } ˆ
x yz i y xz j z x y k idydz
ABEF
=
2
( x yz)
dydz =
ABEF
S.No.
Surface Outward normal
ds
1 OABC – k dx dy z = 0
2 DEFG k dx dy z = c
3 OAFG – j dx dz y = 0
4 BCDE j dx dz y = b
5 ABEF i dy dz x = a
6 OCDG – i dy dz x = 0
b c
b
2
c
2
2 yz
dy ( a yz)
d z = dy
a z
2
0 0
0 0