vector

Vectors 403
EXERCISE 5.8
1. If r = xi yj zk
r
and r = | r | , show that (i) div
3
| r |
= 0,
(ii) div (grad r n ) = n (n + 1) r n – 2 (AMIETE, June 2010) (iii) div (r ) = 3 + r grad .
2. Show that the vector V = ( x3 y) i ( y 3 z) j ( x 2 z)
k is solenoidal.
(R.G.P.V., Bhopal, Dec. 2003)
3. Show that .( A) = .A + (.A)
4. If , , z are cylindrical coordinates, show that grad (log ) and grad are solenoidal vectors.
5. Obtain the expression for 2 f in spherical coordinates from their corresponding expression in
orthogonal curvilinear coordinates.
Prove the following:
6. .( F) ( ). F ( . F )
7. (a) .() = 2
(b)
( A
R) (2 n) A n( A. R)
R
, | |
n n n 2 r R
r r r
8. div ( f g) – div (g f) = f 2 g – g 2 f
5.31 CURL (U.P., I semester, Dec. 2006)
The curl of a vector point function F is defined as below
Curl F
curl F = F
= i j k ( F1 i F2 j F3
k)
x y z
i j k
=
1 2 3
is a vector quantity.
5.32 PHYSICAL MEANING OF CURL
( F F i F j F k)
1 2 3
F
3 F2
F
3 F1
F2 F1
i j k
x y z
y z
x z
x y
F F F
(M.D.U., Dec. 2009, U.P. I Semester, Winter 2009, 2000)
We know that V r,
where is the angular velocity, V is the linear velocity and
r
is the position vector of a point on the rotating body.
Curl V 1 i 2 j 3
k
= V
r x i y j zk
= ( r ) = [( 1i 2 j 3
k) ( xi y j zk)]
=
i j k
=
[( 2z 3y) i ( 1z 3x) j ( 1y 2x) k]
1 2 3
x y z
x y z
= i j k 2z 3y i 1z 3x j 1y 2x k
[( ) ( ) ( ) ]