Mathematical Physics Vol 1

Chapter 4. Field theory

116

4. PARABOLOID ( u , v , ϕ )

x = uv cos ϕ , y = uv sin ϕ , z = 1 2 where u ≥ 0 , v ≥ 0 , 0 ≤ ϕ < 2 π i h u = h v = p u 2 + v 2 , h ϕ = uv . x = a ch u cos v , y = a sh u sin v , z = z where u ≥ 0 , 0 ≤ v < 2 π , − ∞ < z < ∞ i h u = h v = a p sh 2 u + sin 2 v , h z = 1 . x = a sh ξ sin η cos ϕ , y = a sh ξ sin η sin ϕ , z = a ch ξ cos η gde je ξ ≥ 0 , 0 ≤ η ≤ π , 0 ≤ ϕ < 2 π i h ξ = h η = a q sh 2 ξ + sin 2 η , h ϕ = a sh ξ sin η . x = a ch ξ cos η cos ϕ , y = a ch ξ cos η sin ϕ , z = a sh ξ sin η where ξ ≥ 0 , − π 2 ≤ η ≤ π 2 , 0 ≤ ϕ < 2 π i h ξ = h η = a q sh 2 ξ + sin 2 η , h ϕ = a ch ξ cos η . x = c sin β cos ϕ , y = c sin α sin β sin ϕ , z = c ch α cos β where 0 ≤ α < ∞ , 0 ≤ β ≤ π , − π < ϕ ≤ π and Lamé coefficients are h α = h β = c q sh 2 + sin 2 β , h ϕ = c sh α sin β . u 2 − v 2

5. ELLIPTICAL-CYLINDRICAL ( u , v , z )

6. SPHEROID ( ξ , η , ϕ ) a)

b)

7. ELLIPSOIDAL ( α , β , ϕ )

8. ELLIPSOID ( λ , µ , ν )

x 2 a 2 − λ x 2 a 2 − µ x 2 a 2 − ν

y 2 b 2 − λ y 2 b 2 − µ y 2 b 2 − ν

z 2 c 2 − λ z 2 c 2 − µ z 2 c 2 − ν

λ < c 2 < b 2 < a 2

= 1

+

+

c 2 < µ < b 2 < a 2

= 1

+

+

c 2 < b 2 < ν < a 2 ,

= 1

+

+