Mathematical Physics Vol 1

Chapter 4. Field theory

208

Exercise 163 Calculate the following in cylindrical cooridnates a) ∇Φ , b) ∇ · A , c) ∇ × A , d) ∇ 2 Φ .

Solution For cylindrical coordinates ( ρ , φ , z ) u 1 = ρ , u 2 = φ ,

u 3 = z ; e 3 = e z ;

e 1 = e ρ ,

e 2 = e φ ,

h 2 = h φ = ρ ,

h 1 = h ρ = 1 ,

h 3 = h z = 1 ,

and thus A = A ρ e 1 + A φ e 2 + A z e 3 . a)

∂ Φ ∂ u 1

∂ Φ ∂ u 2

∂ Φ ∂ u 3

1 h 1

1 h 2

1 h 3

∇Φ =

e 1 +

e 2 +

e 3 =

∂ Φ ∂ρ

∂ Φ ∂φ

∂ Φ ∂ z

1 1

1 ρ

1 1

e ρ +

e φ +

e z =

=

∂ Φ ∂ρ

∂ Φ ∂φ

∂ Φ ∂ z

1 ρ

e ρ +

e φ +

e z .

=

b)

1 h 1 h 2 h 3 1 ( 1 )( ρ )( 1 ) ∂ ∂ u 1

( A 3 h 1 h 2 ) =

∂ ∂ u 2

∂ ∂ u 3

∇ · A =

( A 1 h 2 h 3 )+

( A 2 h 3 h 1 )+

(( 1 )( ρ ) A z ) =

∂ ∂ρ

∂ ∂φ

∂ ∂ z

(( ρ )( 1 ) A ρ )+

(( 1 )( 1 ) A φ )+

=

1 ρ

( ρ A z ) .

∂ A φ ∂φ

∂ ∂ρ

∂ ∂ z

( ρ A ρ )+

=

+

c)

ρ e φ

h 1 e 1 h 2 e 2 h 3 e 3 ∂ ∂ u 1 ∂ ∂ u 2 ∂ ∂ u 3 A 1 h 1 A 2 h 2 A 3 h 3

e ρ

e z

1 h 1 h 2 h 3

1 ρ

∂ ∂ρ ∂ ∂ z A ρ ρ A φ A z ∂ ∂φ

∇ × A =

=

=

1 ρ

( ρ A φ ) e ρ + ρ

∂ A z ∂ρ

e φ +

∂φ

e z .

∂ A ρ

∂ A ρ

∂ A z ∂φ −

∂ ∂ z

∂ ∂ρ

ρ

( ρ A φ ) −

∂ z −

=

Exercise 164 Write the Laplace equation in terms of parabolic coordinates.