Mathematical Physics Vol 1

4.6 Examples

133

4.6.3 Rotor

Exercise 56 For A = x 2 y i − 2 xz j + 2 yz k find rot rot A .

Solution

rot rot A = ∇ × ( ∇ × A )=( 2 x + 2 ) j .

Exercise 57 For A = 2 yz i − x 2 y j + xz 2 k and Φ = 2 x 2 yz 3 find A × ∇Φ .

Solution A × ∇Φ = − ( 6 x 4 y 2 z 2 + 2 x 3 z 5 ) i +( 4 x 2 yz 5 − 12 x 2 y 2 z 3 ) j +( 4 x 2 yz 4 + 4 x 3 y 2 z 3 ) k .

Exercise 58

Show that the field A =( 2 x 2 + 8 xy 2 z ) i +( 3 x 3 y − 3 xy ) j − ( 4 y 2 z 2 + 2 x 3 z ) k is not rotational ( rot A = 0 ) , and that the field B = xyz 2 A is rotational.

Exercise 59 For A = xz 2 i + 2 y j − 3 xz k and B = 3 xz i + 2 yz j − z 2 k find A × ( ∇ × B ) at point ( 1 , − 1 , 2 ) .

Solution A × ( ∇ × B )= 18 i − 12 j + 16 k .

Exercise 60

Find rot V anddiv V for the vector field V = − r r

x i − y j p x 2 + y 2

= −

.