Mathematical Physics Vol 1

Chapter 4. Field theory

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11. ELLIPTICAL ( λ , µ , z )

x = c λµ , y = c q ( λ 2 − 1 )( 1 − µ 2 ) , z = z Lamé (metric) coefficients are h λ = c s λ 2 − µ 2 λ 2 − 1 , h µ = c s λ 2 − µ 2 1 − µ 2 , h z = 1 .

12. TOROIDAL ( α , β , ϕ )

c sh α cos ϕ ch α − cos β c sin β ch α − cos β gde je: 0 ≤ α < ∞ , − π < β ≤ π , − π < ϕ ≤ π while Lamé coefficients are h α = h β = c ch α − cos β , h ϕ = c sh α ch α − cos β . , y = c sh α sin ϕ ch α − cos β , z =

x =

14. SPHEROIDAL – a ( λ , µ , ϕ ) a)

x = c λµ , y = c q ( λ 2 − 1 )( 1 − µ 2 ) cos ϕ , z = c q ( λ 2 − 1 )( 1 − µ 2 ) sin ϕ , gde je: λ ≤ 1 , − 1 ≤ µ ≤ 1 , 0 ≤ ϕ ≤ 2 π where Lamé coefficients are h λ = c s λ 2 − µ 2 λ 2 − 1 , h µ = c s λ 2 − µ 2 1 − µ 2 , h ϕ = c q ( λ 2 − 1 )( 1 − µ 2 ) . x = c λµ sin ϕ , y = c q ( λ 2 − 1 )( 1 − µ 2 ) , z = c λµ cos ϕ , where Lamé coefficients are µ 2 λ 2 − 1 , h µ = c s λ 2 − µ 2 1 − µ 2 , h ϕ = c λµ .

b)