vector

404 Vectors
=
i j k
x y z
z y xz y x
2 3 3 1 1 2
= ( 1 2) i ( 2 2) j ( 3 3)
k
= 2( 1 i 2 j 3
k) 2
Curl V = 2 which shows that curl of a vector field is connected with rotational properties
of the vector field and justifies the name rotation used for curl.
If Curl F = 0, the field F is termed as irrotational.
2 2 2
Example 41. Find the divergence and curl of v ( xyz) i (3 x y) j ( xz y z)
k at
(2, –1, 1) (Nagpur University, Summer 2003)
Solution. Here, we have
2 2 2
v = ( xyz) i (3 x y) j ( xz y z)
k
Div.
v =
Div 2 2 2
v = ( xyz) (3 x y) ( xz y z)
x y z
= yz + 3x 2 + 2x z – y 2 = –1 + 12 + 4 – 1 = 14 at (2, –1, 1)
Curl v =
i j k
x y z
2 2 2
xyz 3x y xz y z
2
= 2 yz i ( xy z ) j (6 xy xz)
k
Curl at (2, –1, 1)
= 2( 1)(1) i {(2) ( 1) 1} j {6(2)( 1) 2(1)} k
Example 42. If
=
2
2 yz i ( z xy) j (6 xy xz)
k
= 2i 3 j 14k
Ans.
xi yj
zk
V
x y z
2 2 2 ,
find the value of curl V .
Solution. Curl V = V
xi yj
zk
= i j k
x y z 2 2 2 1/2
( x y z )
=
i j k
x y z
x y z
(U.P., I Semester, Winter 2000)
2 2 2 1/2 2 2 2 1/2 2 2 2 1/2
( x y z ) ( x y z ) ( x y z )