Three-dimensional coordinate system
Ellipsoidal coordinates are a three-dimensionalorthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} elliptic coordinate system . Unlike most three-dimensional orthogonal coordinates that featurequadratic coordinate surfaces, the ellipsoidal coordinate system is based onconfocal quadrics .
The Cartesian coordinates( x , y , z ) {\displaystyle (x,y,z)} ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )}
x 2 = ( a 2 + λ ) ( a 2 + μ ) ( a 2 + ν ) ( a 2 − b 2 ) ( a 2 − c 2 ) {\displaystyle x^{2}={\frac {\left(a^{2}+\lambda \right)\left(a^{2}+\mu \right)\left(a^{2}+\nu \right)}{\left(a^{2}-b^{2}\right)\left(a^{2}-c^{2}\right)}}} y 2 = ( b 2 + λ ) ( b 2 + μ ) ( b 2 + ν ) ( b 2 − a 2 ) ( b 2 − c 2 ) {\displaystyle y^{2}={\frac {\left(b^{2}+\lambda \right)\left(b^{2}+\mu \right)\left(b^{2}+\nu \right)}{\left(b^{2}-a^{2}\right)\left(b^{2}-c^{2}\right)}}} z 2 = ( c 2 + λ ) ( c 2 + μ ) ( c 2 + ν ) ( c 2 − b 2 ) ( c 2 − a 2 ) {\displaystyle z^{2}={\frac {\left(c^{2}+\lambda \right)\left(c^{2}+\mu \right)\left(c^{2}+\nu \right)}{\left(c^{2}-b^{2}\right)\left(c^{2}-a^{2}\right)}}} where the following limits apply to the coordinates
− λ < c 2 < − μ < b 2 < − ν < a 2 . {\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}.} Consequently, surfaces of constantλ {\displaystyle \lambda } ellipsoids
x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 , {\displaystyle {\frac {x^{2}}{a^{2}+\lambda }}+{\frac {y^{2}}{b^{2}+\lambda }}+{\frac {z^{2}}{c^{2}+\lambda }}=1,} whereas surfaces of constantμ {\displaystyle \mu } hyperboloids of one sheet
x 2 a 2 + μ + y 2 b 2 + μ + z 2 c 2 + μ = 1 , {\displaystyle {\frac {x^{2}}{a^{2}+\mu }}+{\frac {y^{2}}{b^{2}+\mu }}+{\frac {z^{2}}{c^{2}+\mu }}=1,} because the last term in the lhs is negative, and surfaces of constantν {\displaystyle \nu } hyperboloids of two sheets
x 2 a 2 + ν + y 2 b 2 + ν + z 2 c 2 + ν = 1 {\displaystyle {\frac {x^{2}}{a^{2}+\nu }}+{\frac {y^{2}}{b^{2}+\nu }}+{\frac {z^{2}}{c^{2}+\nu }}=1} because the last two terms in the lhs are negative.
The orthogonal system of quadrics used for the ellipsoidal coordinates areconfocal quadrics .
Scale factors and differential operators [ edit ] For brevity in the equations below, we introduce a function
S ( σ ) = d e f ( a 2 + σ ) ( b 2 + σ ) ( c 2 + σ ) {\displaystyle S(\sigma )\ {\stackrel {\mathrm {def} }{=}}\ \left(a^{2}+\sigma \right)\left(b^{2}+\sigma \right)\left(c^{2}+\sigma \right)} whereσ {\displaystyle \sigma } ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} scale factors can be written
h λ = 1 2 ( λ − μ ) ( λ − ν ) S ( λ ) {\displaystyle h_{\lambda }={\frac {1}{2}}{\sqrt {\frac {\left(\lambda -\mu \right)\left(\lambda -\nu \right)}{S(\lambda )}}}} h μ = 1 2 ( μ − λ ) ( μ − ν ) S ( μ ) {\displaystyle h_{\mu }={\frac {1}{2}}{\sqrt {\frac {\left(\mu -\lambda \right)\left(\mu -\nu \right)}{S(\mu )}}}} h ν = 1 2 ( ν − λ ) ( ν − μ ) S ( ν ) {\displaystyle h_{\nu }={\frac {1}{2}}{\sqrt {\frac {\left(\nu -\lambda \right)\left(\nu -\mu \right)}{S(\nu )}}}} Hence, the infinitesimal volume element equals
d V = ( λ − μ ) ( λ − ν ) ( μ − ν ) 8 − S ( λ ) S ( μ ) S ( ν ) d λ d μ d ν {\displaystyle dV={\frac {\left(\lambda -\mu \right)\left(\lambda -\nu \right)\left(\mu -\nu \right)}{8{\sqrt {-S(\lambda )S(\mu )S(\nu )}}}}\,d\lambda \,d\mu \,d\nu } and theLaplacian is defined by
∇ 2 Φ = 4 S ( λ ) ( λ − μ ) ( λ − ν ) ∂ ∂ λ [ S ( λ ) ∂ Φ ∂ λ ] + 4 S ( μ ) ( μ − λ ) ( μ − ν ) ∂ ∂ μ [ S ( μ ) ∂ Φ ∂ μ ] + 4 S ( ν ) ( ν − λ ) ( ν − μ ) ∂ ∂ ν [ S ( ν ) ∂ Φ ∂ ν ] {\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={}&{\frac {4{\sqrt {S(\lambda )}}}{\left(\lambda -\mu \right)\left(\lambda -\nu \right)}}{\frac {\partial }{\partial \lambda }}\left[{\sqrt {S(\lambda )}}{\frac {\partial \Phi }{\partial \lambda }}\right]\\[1ex]&+{\frac {4{\sqrt {S(\mu )}}}{\left(\mu -\lambda \right)\left(\mu -\nu \right)}}{\frac {\partial }{\partial \mu }}\left[{\sqrt {S(\mu )}}{\frac {\partial \Phi }{\partial \mu }}\right]\\[1ex]&+{\frac {4{\sqrt {S(\nu )}}}{\left(\nu -\lambda \right)\left(\nu -\mu \right)}}{\frac {\partial }{\partial \nu }}\left[{\sqrt {S(\nu )}}{\frac {\partial \Phi }{\partial \nu }}\right]\end{aligned}}} Other differential operators such as∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } ∇ × F {\displaystyle \nabla \times \mathbf {F} } ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )}
Angular parametrization [ edit ] An alternative (but non-orthogonal) parametrization exists that closely follows the angular parametrization ofspherical coordinates :[ 1]
x = a s sin θ cos ϕ , {\displaystyle x=as\sin \theta \cos \phi ,} y = b s sin θ sin ϕ , {\displaystyle y=bs\sin \theta \sin \phi ,} z = c s cos θ . {\displaystyle z=cs\cos \theta .} Here,s > 0 {\displaystyle s>0} θ ∈ [ 0 , π ] {\displaystyle \theta \in [0,\pi ]} ϕ ∈ [ 0 , 2 π ] {\displaystyle \phi \in [0,2\pi ]}
d x d y d z = a b c s 2 sin θ d s d θ d ϕ . {\displaystyle dx\,dy\,dz=abc\,s^{2}\sin \theta \,ds\,d\theta \,d\phi .} Morse PM, Feshbach H (1953).Methods of Theoretical Physics, Part I . New York: McGraw-Hill. p. 663. Zwillinger D (1992).Handbook of Integration . Boston, MA: Jones and Bartlett. p. 114.ISBN 0-86720-293-9 Sauer R, Szabó I (1967).Mathematische Hilfsmittel des Ingenieurs . New York: Springer Verlag. pp. 101– 102.LCCN 67025285 . Korn GA,Korn TM (1961).Mathematical Handbook for Scientists and Engineers 176 .LCCN 59014456 . Margenau H, Murphy GM (1956).The Mathematics of Physics and Chemistry 178 –180.LCCN 55010911 . Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions 40 –44 (Table 1.10).ISBN 0-387-02732-7 Landau LD, Lifshitz EM, Pitaevskii LP (1984).Electrodynamics of Continuous Media (Volume 8 of theCourse of Theoretical Physics ) (2nd ed.). New York: Pergamon Press. pp. 19– 29.ISBN 978-0-7506-2634-7
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