Mathematical Physics Vol 1

Chapter 4. Field theory

134

Exercise 61 Show that rot rot A = − ∆ A + grad div A .

Exercise 62 Determine the velocity rotor of an arbitrary point on a rigid body rotating around a fixed pole.

Solution The velocity of a point on a body rotating around a fixed pole (which can be assumed to be the origin of the coordinate system, without loss of generality) is given by the relation = i ( ω y z − ω z y )+ j ( ω z x − ω x z )+ k ( ω x y − ω y x ) , where ω is the angular velocity, and ρ the position vector of the point on the body. Given that the angular velocity of a body is equal for all points on this body, it does not depend on the coordinates x , y and z , and it follows that ∂ ∂ x ( ˙ y )= ∂ ∂ x ( ω z x − ω x z )= ω z ∂ ∂ y ( ˙ x )= ∂ ∂ y ( ω y z − ω z y )= − ω z , ∂ ∂ z ( ˙ x )= ∂ ∂ z ( ω y z − ω z y )= ω y ∂ ∂ x ( ˙ z )= ∂ ∂ x ( ω x y − ω y x )= − ω y , ∂ ∂ y ( ˙ z )= ∂ ∂ y ( ω x y − ω y z )= ω x ∂ ∂ z ( ˙ y )= ∂ ∂ z ( ω z x − ω x z )= − ω x . Thus, the velocity rotor v , according to (4.46) is v = ω × ρ = i j k ω x ω y ω z x y z

i

j

k

= i

∂ ˙ y ∂ z

+ j

∂ ˙ z ∂ x

+ k

∂ ˙ x ∂ y

∂ ˙ z ∂ y −

∂ ˙ x ∂ z −

∂ ˙ y ∂ x −

∂ ∂ x

∂ ∂ y

∂ ∂ z

rot v =

.

˙ x

˙ y

˙ z

From these relations we finally obtain

rot v = i [ ω x − ( − ω x )]+ j [ ω y − ( − ω y )]+ k [ ω z − ( − ω z )]= 2 ω .

Exercise 63 If V = xz 3 i − 2 x 2 yz j + 2 yz 4 k , find ∇ × V at point A ( 1 , − 1 , 1 ) .