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Ellipsoidal coordinates
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<h2 id="basic-formulae">Basic formulae</h2> <p>The Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be produced from the ellipsoidal coordinates ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} by the equations </p> 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)}}} <p>where the following limits apply to the coordinates </p> − λ < c 2 < − μ < b 2 < − ν < a 2 . {\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}.} <p>Consequently, surfaces of constant λ {\displaystyle \lambda } are <a href="/facts/Ellipsoid/pe9vm0Sh">ellipsoids</a> </p> 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,} <p>whereas surfaces of constant μ {\displaystyle \mu } are <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloids</a> of one sheet </p> 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,} <p>because the last term in the lhs is negative, and surfaces of constant ν {\displaystyle \nu } are <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloids</a> of two sheets </p> 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} <p>because the last two terms in the lhs are negative. </p><p>The orthogonal system of quadrics used for the ellipsoidal coordinates are <a href="/facts/Confocal_quadrics/sNANGYBF">confocal quadrics</a>. </p> <h2 id="scale-factors-and-differential-operators">Scale factors and differential operators</h2> <p>For brevity in the equations below, we introduce a function </p> 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)} <p>where σ {\displaystyle \sigma } can represent any of the three variables ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} . Using this function, the <a href="/facts/Curvilinear_coordinates/RQD7Cajd">scale factors</a> can be written </p> 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 )}}}} <p>Hence, the infinitesimal volume element equals </p> 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 } <p>and the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a> is defined by </p> ∇ 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}}} <p>Other differential operators such as ∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } and ∇ × F {\displaystyle \nabla \times \mathbf {F} } can be expressed in the coordinates ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} by substituting the scale factors into the general formulae found in <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal coordinates</a>. </p> <h2 id="angular-parametrization">Angular parametrization</h2> <p>An alternative parametrization exists that closely follows the angular parametrization of <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> 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 .} <p>Here, s > 0 {\displaystyle s>0} parametrizes the concentric ellipsoids around the origin and θ ∈ [ 0 , π ] {\displaystyle \theta \in [0,\pi ]} and ϕ ∈ [ 0 , 2 π ] {\displaystyle \phi \in [0,2\pi ]} are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is </p> 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 .} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Ellipsoidal_latitude/I4kBSdYu">Ellipsoidal latitude</a></li> <li><a href="/facts/Focaloid/Exz67YAT">Focaloid</a> (shell given by two coordinate surfaces)</li> <li><a href="/facts/Map_projection_of_the_triaxial_ellipsoid/zPc9Hl10">Map projection of the triaxial ellipsoid</a></li></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Morse PM, Feshbach H (1953). <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill. p. 663.</li> <li>Zwillinger D (1992). <i>Handbook of Integration</i>. Boston, MA: Jones and Bartlett. p. 114. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-86720-293-9.</li> <li>Sauer R, Szabó I (1967). <i>Mathematische Hilfsmittel des Ingenieurs</i>. New York: Springer Verlag. pp. 101–102. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/67025285">67025285</a>.</li> <li>Korn GA, <a href="/facts/Theresa_M._Korn/qKomWYbN">Korn TM</a> (1961). <a href="https://archive.org/details/mathematicalhand0000korn"><i>Mathematical Handbook for Scientists and Engineers</i></a>. New York: McGraw-Hill. p. <a href="https://archive.org/details/mathematicalhand0000korn/page/176">176</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/59014456">59014456</a>.</li> <li>Margenau H, Murphy GM (1956). <a href="https://archive.org/details/mathematicsphysi00marg_500"><i>The Mathematics of Physics and Chemistry</i></a>. New York: D. van Nostrand. pp. <a href="https://archive.org/details/mathematicsphysi00marg_500/page/n191">178</a>–180. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/55010911">55010911</a>.</li> <li>Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)". <a href="https://archive.org/details/fieldtheoryhandb00moon"><i>Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions</i></a> (corrected 2nd, 3rd print ed.). New York: Springer Verlag. pp. <a href="https://archive.org/details/fieldtheoryhandb00moon/page/n42">40</a>–44 (Table 1.10). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-02732-7.</li></ul> <h3>Unusual convention</h3> <ul><li>Landau LD, Lifshitz EM, Pitaevskii LP (1984). <i>Electrodynamics of Continuous Media (Volume 8 of the <a href="/facts/Course_of_Theoretical_Physics/Luk16l8o">Course of Theoretical Physics</a>)</i> (2nd ed.). New York: Pergamon Press. pp. 19–29. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7506-2634-7. Uses (ξ, η, ζ) coordinates that have the units of distance squared.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html">MathWorld description of confocal ellipsoidal coordinates</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Ellipsoid Quadrupole Moment". 9 October 2013. <a href="https://photonics101.com/multipole-moments-electric/quadrupole-multipole-moments-homogeneously-charged-ellipsoid#hints" target="_blank">https://photonics101.com/multipole-moments-electric/quadrupole-multipole-moments-homogeneously-charged-ellipsoid#hints</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>