vector

418 Vectors
i j k
( a r )
n
= x y z
r
azay ax az ay
ax
2 3 3 1 1 2
n n n
r r r
ay 1 ax 2 ax 3 az 1
ay 1 ax
2 az 2 ay
3
= i j
n
n n n
y r z
r x r z
r
ax 3 az 1 az 2 ay 3
k
n
n
x
r y
r
Now, r 2 = x 2 + y 2 + z 2
r
r
x
2r
= 2x
x
x
r
Similarly,
r
y
= ,
r
z
y
r
z
r
a r
n
1
y
1
= i . ( n
r
nr a1y a2x)
a
n 1
r
r
n
1
z
1
nr ( ax 3 az 1 ) ( a1
)
r
n + two similar terms
r
=
n 2 a1 n
2 a1
i ( a
2 1y a2xy) ( axz
2 3 a1z
)
n
n n
n
r r r r
+ two similar terms
=
2a1
n 2 2 n
i a
2 1( y z ) ( axy
2 2 a3xz)
n n
n
+ two similar terms
r r r
Adding and subtracting
n 2
ax to third and from second term, we get
n 2
r
1
a r
2a1 na1
2 2 2 n 2
n
r
= i ( x y z ) ( ax
2 2 1 axy 2 a3xz)
n n
n
r r r
+ two similar terms
2a1 na1
2 n
= i r xax (
2 2 1 a2y a3
z)
n n
n
+ two similar terms
r r r
2a1 na1
n
2a2 na2
n
= i x( ax
2 1 ay 2 az 3 )
n n n
j
y( a
2 2 y az 3 ax 1 )
n n n
r r r
r r r
2a3 na3
n
k
z( az
2 3 ax 1 a2
y)
n n n
r r r
2 n n
= ( a
n 1 i a2 j a3k )
n ( a1 i a2 j a3k
) ( 2 1 2 3 )(
ax a y az xi yj
zk )
n
r
r
r
2 n n
= ( a1 i a2 j a3k) ( ax
2 1 a2y az 3 )( xi yj
zk)
n
n
r
r
2 n n
= a ( a. r)
r
n n
2
Proved.
r r
Example 63. If f and g are two scalar point functions, prove that
div (f g) = f 2 g + f g. (U.P., I Semester, compartment, Winter 2001)