Mathematical Physics Vol 1

Chapter 4. Field theory

218

Solution

∂ r ∂ρ ∂ r ∂φ ∂ r ∂ z

x i + y j p x 2 + y 2

= cos φ i + sin φ j ,

∇ ρ =

= cos φ i + sin φ j ,

a)

sin φ i + cos φ j ρ ,

= − ρ sin φ i + ρ cos φ j , ∇ φ = −

∇ z = k

= k ,

∂ r ∂ r = sin θ cos φ i + sin θ sin φ j + cos θ k ∂ r ∂θ = r cos θ cos φ i + r cos θ sin φ j − r sin θ k

b)

∂ r ∂φ

= − r sin θ sin φ i + r sin θ cos φ j

x i + y j + z k p x 2 + y 2 + z 2 xz i + yz j − p x 2 + y 2 k ( x 2 + y 2 + z 2 ) p x 2 + y 2 y i + x j = −

∇ r =

= sin θ cos φ i + sin θ sin φ j + cos θ k

= cos θ cos φ i + cos θ sin φ j − sin θ k r

∇ θ =

sin φ + cos φ r sin θ

∇ φ = −

x 2 + y 2

∂ r ∂ u ∂ r ∂ v ∂ r ∂ z

u i + v j u 2 + v 2

∇ u =

= u i + v j ,

c)

v i + u j

∇ v = −

= − v i + u j ,

,

u 2 + v 2

∇ z = k .

= k ,

Exercise 180 Express the equation

∂ 2 φ ∂ x 2

∂ 2 φ ∂ y 2

= φ in terms of elliptic cylindrical coordinates.

+

Solution

∂ 2 φ ∂ u 2

∂ 2 φ ∂ v 2

= a 2 ( sh 2 u + sin 2 v ) φ

+

Exercise 181 Express Schrodinger’s equation (quantum mechanics) ∇ 2 ψ + 8 π 2 m h ( e − v ( x , y , z ))= 0 .