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Prolate spheroidal coordinates
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<h2 id="definition">Definition</h2> <p>The most common definition of prolate spheroidal coordinates ( μ , ν , φ ) {\displaystyle (\mu ,\nu ,\varphi )} is </p> x = a sinh μ sin ν cos φ {\displaystyle x=a\sinh \mu \sin \nu \cos \varphi } y = a sinh μ sin ν sin φ {\displaystyle y=a\sinh \mu \sin \nu \sin \varphi } z = a cosh μ cos ν {\displaystyle z=a\cosh \mu \cos \nu } <p>where μ {\displaystyle \mu } is a nonnegative real number and ν ∈ [ 0 , π ] {\displaystyle \nu \in [0,\pi ]} . The azimuthal angle φ {\displaystyle \varphi } belongs to the interval [ 0 , 2 π ] {\displaystyle [0,2\pi ]} . </p><p>The trigonometric identity </p> z 2 a 2 cosh 2 μ + x 2 + y 2 a 2 sinh 2 μ = cos 2 ν + sin 2 ν = 1 {\displaystyle {\frac {z^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {x^{2}+y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1} <p>shows that surfaces of constant μ {\displaystyle \mu } form <a href="/facts/Prolate/GJyyLxQ7">prolate</a> <a href="/facts/Spheroid/GJyyLxQ7">spheroids</a>, since they are <a href="/facts/Ellipse/1wUDnGxY">ellipses</a> rotated about the axis joining their foci. Similarly, the hyperbolic trigonometric identity </p> z 2 a 2 cos 2 ν − x 2 + y 2 a 2 sin 2 ν = cosh 2 μ − sinh 2 μ = 1 {\displaystyle {\frac {z^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {x^{2}+y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1} <p>shows that surfaces of constant ν {\displaystyle \nu } form <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloids</a> of revolution. </p><p>The distances from the foci located at ( x , y , z ) = ( 0 , 0 , ± a ) {\displaystyle (x,y,z)=(0,0,\pm a)} are </p> r ± = x 2 + y 2 + ( z ∓ a ) 2 = a ( cosh μ ∓ cos ν ) . {\displaystyle r_{\pm }={\sqrt {x^{2}+y^{2}+(z\mp a)^{2}}}=a(\cosh \mu \mp \cos \nu ).} <h2 id="scale-factors">Scale factors</h2> <p>The scale factors for the elliptic coordinates ( μ , ν ) {\displaystyle (\mu ,\nu )} are equal </p> h μ = h ν = a sinh 2 μ + sin 2 ν {\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}} <p>whereas the azimuthal scale factor is </p> h φ = a sinh μ sin ν , {\displaystyle h_{\varphi }=a\sinh \mu \sin \nu ,} <p>resulting in a metric of </p> d s 2 = h μ 2 d μ 2 + h ν 2 d ν 2 + h φ 2 d φ 2 = a 2 [ ( sinh 2 μ + sin 2 ν ) d μ 2 + ( sinh 2 μ + sin 2 ν ) d ν 2 + ( sinh 2 μ sin 2 ν ) d φ 2 ] . {\displaystyle {\begin{aligned}ds^{2}&=h_{\mu }^{2}d\mu ^{2}+h_{\nu }^{2}d\nu ^{2}+h_{\varphi }^{2}d\varphi ^{2}\\&=a^{2}\left[(\sinh ^{2}\mu +\sin ^{2}\nu )d\mu ^{2}+(\sinh ^{2}\mu +\sin ^{2}\nu )d\nu ^{2}+(\sinh ^{2}\mu \sin ^{2}\nu )d\varphi ^{2}\right].\end{aligned}}} <p>Consequently, an infinitesimal volume element equals </p> d V = a 3 sinh μ sin ν ( sinh 2 μ + sin 2 ν ) d μ d ν d φ {\displaystyle dV=a^{3}\sinh \mu \sin \nu (\sinh ^{2}\mu +\sin ^{2}\nu )\,d\mu \,d\nu \,d\varphi } <p>and the Laplacian can be written </p> ∇ 2 Φ = 1 a 2 ( sinh 2 μ + sin 2 ν ) [ ∂ 2 Φ ∂ μ 2 + ∂ 2 Φ ∂ ν 2 + coth μ ∂ Φ ∂ μ + cot ν ∂ Φ ∂ ν ] + 1 a 2 sinh 2 μ sin 2 ν ∂ 2 Φ ∂ φ 2 {\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={}&{\frac {1}{a^{2}(\sinh ^{2}\mu +\sin ^{2}\nu )}}\left[{\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}+\coth \mu {\frac {\partial \Phi }{\partial \mu }}+\cot \nu {\frac {\partial \Phi }{\partial \nu }}\right]\\[6pt]&{}+{\frac {1}{a^{2}\sinh ^{2}\mu \sin ^{2}\nu }}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}\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 ( μ , ν , φ ) {\displaystyle (\mu ,\nu ,\varphi )} by substituting the scale factors into the general formulae found in <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal coordinates</a>. </p> <h2 id="alternative-definition">Alternative definition</h2> <p>An alternative and geometrically intuitive set of prolate spheroidal coordinates ( σ , τ , ϕ ) {\displaystyle (\sigma ,\tau ,\phi )} are sometimes used, where σ = cosh μ {\displaystyle \sigma =\cosh \mu } and τ = cos ν {\displaystyle \tau =\cos \nu } . Hence, the curves of constant σ {\displaystyle \sigma } are prolate spheroids, whereas the curves of constant τ {\displaystyle \tau } are hyperboloids of revolution. The coordinate τ {\displaystyle \tau } belongs to the interval [−1, 1], whereas the σ {\displaystyle \sigma } coordinate must be greater than or equal to one. </p><p>The coordinates σ {\displaystyle \sigma } and τ {\displaystyle \tau } have a simple relation to the distances to the foci F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} . For any point in the plane, the <i>sum</i> d 1 + d 2 {\displaystyle d_{1}+d_{2}} of its distances to the foci equals 2 a σ {\displaystyle 2a\sigma } , whereas their <i>difference</i> d 1 − d 2 {\displaystyle d_{1}-d_{2}} equals 2 a τ {\displaystyle 2a\tau } . Thus, the distance to F 1 {\displaystyle F_{1}} is a ( σ + τ ) {\displaystyle a(\sigma +\tau )} , whereas the distance to F 2 {\displaystyle F_{2}} is a ( σ − τ ) {\displaystyle a(\sigma -\tau )} . (Recall that F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are located at z = − a {\displaystyle z=-a} and z = + a {\displaystyle z=+a} , respectively.) This gives the following expressions for σ {\displaystyle \sigma } , τ {\displaystyle \tau } , and φ {\displaystyle \varphi } : </p> σ = 1 2 a ( x 2 + y 2 + ( z + a ) 2 + x 2 + y 2 + ( z − a ) 2 ) {\displaystyle \sigma ={\frac {1}{2a}}\left({\sqrt {x^{2}+y^{2}+(z+a)^{2}}}+{\sqrt {x^{2}+y^{2}+(z-a)^{2}}}\right)} τ = 1 2 a ( x 2 + y 2 + ( z + a ) 2 − x 2 + y 2 + ( z − a ) 2 ) {\displaystyle \tau ={\frac {1}{2a}}\left({\sqrt {x^{2}+y^{2}+(z+a)^{2}}}-{\sqrt {x^{2}+y^{2}+(z-a)^{2}}}\right)} φ = arctan ( y x ) {\displaystyle \varphi =\arctan \left({\frac {y}{x}}\right)} <p>Unlike the analogous <a href="/facts/Oblate_spheroidal_coordinates/AqznWJGK">oblate spheroidal coordinates</a>, the prolate spheroid coordinates (σ, τ, φ) are <i>not</i> degenerate; in other words, there is a <a href="/facts/1-to-1_mapping/j4gTuTmW">unique, reversible correspondence</a> between them and the <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> </p> x = a ( σ 2 − 1 ) ( 1 − τ 2 ) cos φ {\displaystyle x=a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}\cos \varphi } y = a ( σ 2 − 1 ) ( 1 − τ 2 ) sin φ {\displaystyle y=a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}\sin \varphi } z = a σ τ {\displaystyle z=a\ \sigma \ \tau } <h2 id="alternative-scale-factors">Alternative scale factors</h2> <p>The scale factors for the alternative elliptic coordinates ( σ , τ , φ ) {\displaystyle (\sigma ,\tau ,\varphi )} are </p> h σ = a σ 2 − τ 2 σ 2 − 1 {\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}} h τ = a σ 2 − τ 2 1 − τ 2 {\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}} <p>while the azimuthal scale factor is now </p> h φ = a ( σ 2 − 1 ) ( 1 − τ 2 ) {\displaystyle h_{\varphi }=a{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}} <p>Hence, the infinitesimal volume element becomes </p> d V = a 3 ( σ 2 − τ 2 ) d σ d τ d φ {\displaystyle dV=a^{3}(\sigma ^{2}-\tau ^{2})\,d\sigma \,d\tau \,d\varphi } <p>and the Laplacian equals </p> ∇ 2 Φ = 1 a 2 ( σ 2 − τ 2 ) { ∂ ∂ σ [ ( σ 2 − 1 ) ∂ Φ ∂ σ ] + ∂ ∂ τ [ ( 1 − τ 2 ) ∂ Φ ∂ τ ] } + 1 a 2 ( σ 2 − 1 ) ( 1 − τ 2 ) ∂ 2 Φ ∂ φ 2 {\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={}&{\frac {1}{a^{2}(\sigma ^{2}-\tau ^{2})}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}-1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[(1-\tau ^{2}){\frac {\partial \Phi }{\partial \tau }}\right]\right\}\\&{}+{\frac {1}{a^{2}(\sigma ^{2}-1)(1-\tau ^{2})}}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}\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 ( σ , τ ) {\displaystyle (\sigma ,\tau )} by substituting the scale factors into the general formulae found in <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal coordinates</a>. </p><p>As is the case with <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>, <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> may be solved by the method of <a href="/facts/Separation_of_variables/oJdkw5xq">separation of variables</a> to yield solutions in the form of <a href="/facts/Prolate_spheroidal_wave_function/6HEjqT3c">prolate spheroidal harmonics</a>, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968). </p> <h2 id="bibliography">Bibliography</h2> <h3>No angles convention</h3> <ul><li>Morse PM, Feshbach H (1953). <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill. p. 661. Uses <i>ξ</i>1 = <i>a</i> cosh <i>μ</i>, <i>ξ</i>2 = sin <i>ν</i>, and <i>ξ</i>3 = cos <i>φ</i>.</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. Same as Morse & Feshbach (1953), substituting <i>u</i><i>k</i> for <i>ξ</i><i>k</i>.</li> <li>Smythe, WR (1968). <i>Static and Dynamic Electricity</i> (3rd ed.). New York: McGraw-Hill.</li> <li>Sauer R, Szabó I (1967). <i>Mathematische Hilfsmittel des Ingenieurs</i>. New York: Springer Verlag. p. 97. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/67025285">67025285</a>. Uses coordinates <i>ξ</i> = cosh <i>μ</i>, <i>η</i> = sin <i>ν</i>, and <i>φ</i>.</li></ul> <h3>Angle convention</h3> <ul><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/177">177</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/59014456">59014456</a>. Korn and Korn use the (μ, ν, φ) coordinates, but also introduce the degenerate (σ, τ, φ) coordinates.</li> <li>Margenau H, Murphy GM (1956). <a href="https://archive.org/details/mathematicsofphy0002marg"><i>The Mathematics of Physics and Chemistry</i></a>. New York: D. van Nostrand. pp. <a href="https://archive.org/details/mathematicsofphy0002marg/page/180">180</a>–182. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/55010911">55010911</a>. Similar to Korn and Korn (1961), but uses <a href="/facts/Colatitude/NHurcapj">colatitude</a> θ = 90° - ν instead of <a href="/facts/Latitude/IBYqkjhU">latitude</a> ν.</li> <li>Moon PH, Spencer DE (1988). "Prolate Spheroidal Coordinates (η, θ, ψ)". <i>Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions</i> (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 28–30 (Table 1.06). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-02732-7. Moon and Spencer use the colatitude convention <i>θ</i> = 90° − <i>ν</i>, and rename <i>φ</i> as <i>ψ</i>.</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. Treats the prolate spheroidal coordinates as a limiting case of the general <a href="/facts/Ellipsoidal_coordinates/7U3CW9hI">ellipsoidal coordinates</a>. 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/ProlateSpheroidalCoordinates.html">MathWorld description of prolate spheroidal coordinates</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lehtola, Susi (21 May 2019). "A review on non-relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules". Int. J. Quantum Chem. 119 (19): e25968. arXiv:1902.01431. doi:10.1002/qua.25968. <a href="https://doi.org/10.1002%2Fqua.25968" target="_blank">https://doi.org/10.1002%2Fqua.25968</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>