Coordinate surfaces

The surfaces of constant μ form oblate spheroids, by the trigonometric identity

$${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {z^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}$$

since they are ellipses rotated about the z-axis, which separates their foci. An ellipse in the x-z plane (Figure 2) has a major semiaxis of length a cosh μ along the x-axis, whereas its minor semiaxis has length a sinh μ along the z-axis. The foci of all the ellipses in the x-z plane are located on the x-axis at ±a.

Similarly, the surfaces of constant ν form one-sheet half hyperboloids of revolution by the hyperbolic trigonometric identity

$${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {z^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}$$

For positive ν, the half-hyperboloid is above the x-y plane (i.e., has positive z) whereas for negative ν, the half-hyperboloid is below the x-y plane (i.e., has negative z). Geometrically, the angle ν corresponds to the angle of the asymptotes of the hyperbola. The foci of all the hyperbolae are likewise located on the x-axis at ±a.

Inverse transformation

The (μ, ν, φ) coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle φ is given by the formula

$${\displaystyle \tan \phi ={\frac {y}{x}}}$$

The cylindrical radius ρ of the point P is given by

$${\displaystyle \rho ^{2}=x^{2}+y^{2}}$$

and its distances to the foci in the plane defined by φ is given by

$${\displaystyle {\begin{aligned}d_{1}^{2}=(\rho +a)^{2}+z^{2}\\d_{2}^{2}=(\rho -a)^{2}+z^{2}\end{aligned}}}$$

The remaining coordinates μ and ν can be calculated from the equations

$${\displaystyle {\begin{aligned}\cosh \mu &={\frac {d_{1}+d_{2}}{2a}}\\\cos \nu &={\frac {d_{1}-d_{2}}{2a}}\end{aligned}}}$$

where the sign of μ is always non-negative, and the sign of ν is the same as that of z.

Another method to compute the inverse transform is

$${\displaystyle {\begin{aligned}\mu &=\operatorname {Re} \operatorname {arcosh} {\frac {\rho +zi}{a}}\\\nu &=\operatorname {Im} \operatorname {arcosh} {\frac {\rho +zi}{a}}\\\phi &=\arctan {\frac {y}{x}}\end{aligned}}}$$

where

$${\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}}$$

Scale factors

The scale factors for the coordinates μ and ν are equal

$${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}$$

whereas the azimuthal scale factor equals

$${\displaystyle h_{\phi }=a\cosh \mu \ \cos \nu }$$

Consequently, an infinitesimal volume element equals

$${\displaystyle dV=a^{3}\cosh \mu \ \cos \nu \ \left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu \,d\nu \,d\phi }$$

and the Laplacian can be written

$${\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {1}{\cosh \mu }}{\frac {\partial }{\partial \mu }}\left(\cosh \mu {\frac {\partial \Phi }{\partial \mu }}\right)+{\frac {1}{\cos \nu }}{\frac {\partial }{\partial \nu }}\left(\cos \nu {\frac {\partial \Phi }{\partial \nu }}\right)\right]+{\frac {1}{a^{2}\cosh ^{2}\mu \cos ^{2}\nu }}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}$$

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {F} }$ can be expressed in the coordinates (μ, ν, φ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Basis Vectors

The orthonormal basis vectors for the ${\displaystyle \mu ,\nu ,\phi }$ coordinate system can be expressed in Cartesian coordinates as

$${\displaystyle {\begin{aligned}{\hat {e}}_{\mu }&={\frac {1}{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}\left(\sinh \mu \cos \nu \cos \phi {\boldsymbol {\hat {i}}}+\sinh \mu \cos \nu \sin \phi {\boldsymbol {\hat {j}}}+\cosh \mu \sin \nu {\boldsymbol {\hat {k}}}\right)\\{\hat {e}}_{\nu }&={\frac {1}{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}\left(-\cosh \mu \sin \nu \cos \phi {\boldsymbol {\hat {i}}}-\cosh \mu \sin \nu \sin \phi {\boldsymbol {\hat {j}}}+\sinh \mu \cos \nu {\boldsymbol {\hat {k}}}\right)\\{\hat {e}}_{\phi }&=-\sin \phi {\boldsymbol {\hat {i}}}+\cos \phi {\boldsymbol {\hat {j}}}\end{aligned}}}$$

where ${\displaystyle {\boldsymbol {\hat {i}}},{\boldsymbol {\hat {j}}},{\boldsymbol {\hat {k}}}}$ are the Cartesian unit vectors. Here, ${\displaystyle {\hat {e}}_{\mu }}$ is the outward normal vector to the oblate spheroidal surface of constant ${\displaystyle \mu }$, ${\displaystyle {\hat {e}}_{\phi }}$ is the same azimuthal unit vector from spherical coordinates, and ${\displaystyle {\hat {e}}_{\nu }}$ lies in the tangent plane to the oblate spheroid surface and completes the right-handed basis set.

Scale factors

The scale factors for ${\displaystyle (\zeta ,\xi ,\phi )}$ are:

$${\displaystyle {\begin{aligned}h_{\zeta }&=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1+\zeta ^{2}}}}\\h_{\xi }&=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1-\xi ^{2}}}}\\h_{\phi }&=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}\end{aligned}}}$$

Knowing the scale factors, various functions of the coordinates can be calculated by the general method outlined in the orthogonal coordinates article. The infinitesimal volume element is:

$${\displaystyle dV=a^{3}\left(\zeta ^{2}+\xi ^{2}\right)d\zeta \,d\xi \,d\phi }$$

The gradient is:

$${\displaystyle \nabla V={\frac {1}{h_{\zeta }}}{\frac {\partial V}{\partial \zeta }}\,{\hat {\zeta }}+{\frac {1}{h_{\xi }}}{\frac {\partial V}{\partial \xi }}\,{\hat {\xi }}+{\frac {1}{h_{\phi }}}{\frac {\partial V}{\partial \phi }}\,{\hat {\phi }}}$$

The divergence is:

$${\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{a(\zeta ^{2}+\xi ^{2})}}\left\{{\frac {\partial }{\partial \zeta }}\left({\sqrt {1+\zeta ^{2}}}{\sqrt {\zeta ^{2}+\xi ^{2}}}F_{\zeta }\right)+{\frac {\partial }{\partial \xi }}\left({\sqrt {1-\xi ^{2}}}{\sqrt {\zeta ^{2}+\xi ^{2}}}F_{\xi }\right)\right\}+{\frac {1}{a{\sqrt {1+\zeta ^{2}}}{\sqrt {1-\xi ^{2}}}}}{\frac {\partial F_{\phi }}{\partial \phi }}}$$

and the Laplacian equals

$${\displaystyle \nabla ^{2}V={\frac {1}{a^{2}\left(\zeta ^{2}+\xi ^{2}\right)}}\left\{{\frac {\partial }{\partial \zeta }}\left[\left(1+\zeta ^{2}\right){\frac {\partial V}{\partial \zeta }}\right]+{\frac {\partial }{\partial \xi }}\left[\left(1-\xi ^{2}\right){\frac {\partial V}{\partial \xi }}\right]\right\}+{\frac {1}{a^{2}\left(1+\zeta ^{2}\right)\left(1-\xi ^{2}\right)}}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}}$$

Oblate spheroidal harmonics

As is the case with spherical coordinates and spherical harmonics, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate.

Following the technique of separation of variables, a solution to Laplace's equation is written:

$${\displaystyle V=Z(\zeta )\,\Xi (\xi )\,\Phi (\phi )}$$

This yields three separate differential equations in each of the variables:

$${\displaystyle {\begin{aligned}{\frac {d}{d\zeta }}\left[(1+\zeta ^{2}){\frac {dZ}{d\zeta }}\right]+{\frac {m^{2}Z}{1+\zeta ^{2}}}-n(n+1)Z=0\\{\frac {d}{d\xi }}\left[(1-\xi ^{2}){\frac {d\Xi }{d\xi }}\right]-{\frac {m^{2}\Xi }{1-\xi ^{2}}}+n(n+1)\Xi =0\\{\frac {d^{2}\Phi }{d\phi ^{2}}}=-m^{2}\Phi \end{aligned}}}$$

where m is a constant which is an integer because the φ variable is periodic with period 2π. n will then be an integer. The solution to these equations are:

$${\displaystyle {\begin{aligned}Z_{mn}&=A_{1}P_{n}^{m}(i\zeta )+A_{2}Q_{n}^{m}(i\zeta )\\[1ex]\Xi _{mn}&=A_{3}P_{n}^{m}(\xi )+A_{4}Q_{n}^{m}(\xi )\\[1ex]\Phi _{m}&=A_{5}e^{im\phi }+A_{6}e^{-im\phi }\end{aligned}}}$$

where the ${\displaystyle A_{i}}$ are constants and ${\displaystyle P_{n}^{m}(z)}$ and ${\displaystyle Q_{n}^{m}(z)}$ are associated Legendre polynomials of the first and second kind respectively. The product of the three solutions is called an oblate spheroidal harmonic and the general solution to Laplace's equation is written:

$${\displaystyle V=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\,Z_{mn}(\zeta )\,\Xi _{mn}(\xi )\,\Phi _{m}(\phi )}$$

The constants will combine to yield only four independent constants for each harmonic.

Coordinate surfaces

Similar to its counterpart μ, the surfaces of constant σ form oblate spheroids

$${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\sigma ^{2}}}+{\frac {z^{2}}{a^{2}\left(\sigma ^{2}-1\right)}}=1}$$

Similarly, the surfaces of constant τ form full one-sheet hyperboloids of revolution

$${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\tau ^{2}}}-{\frac {z^{2}}{a^{2}\left(1-\tau ^{2}\right)}}=1}$$

Scale factors

The scale factors for the alternative oblate spheroidal coordinates ${\displaystyle (\sigma ,\tau ,\phi )}$ are

$${\displaystyle {\begin{aligned}h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}\\h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}\end{aligned}}}$$

whereas the azimuthal scale factor is ${\displaystyle h_{\phi }=a\sigma \tau }$.

Hence, the infinitesimal volume element can be written

$${\displaystyle dV=a^{3}\sigma \tau {\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}\,d\sigma \,d\tau \,d\phi }$$

and the Laplacian equals

$${\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left\{{\frac {\sqrt {\sigma ^{2}-1}}{\sigma }}{\frac {\partial }{\partial \sigma }}\left[\left(\sigma {\sqrt {\sigma ^{2}-1}}\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\sqrt {1-\tau ^{2}}}{\tau }}{\frac {\partial }{\partial \tau }}\left[\left(\tau {\sqrt {1-\tau ^{2}}}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}$$

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {F} }$ can be expressed in the coordinates ${\displaystyle (\sigma ,\tau )}$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

As is the case with spherical coordinates, Laplaces equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate (See Smythe, 1968).

No angles convention

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 662. Uses ξ₁ = a sinh μ, ξ₂ = sin ν, and ξ₃ = cos φ.
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 115. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• Smythe, WR (1968). Static and Dynamic Electricity (3rd ed.). New York: McGraw-Hill.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 98. LCCN 67025285. Uses hybrid coordinates ξ = sinh μ, η = sin ν, and φ.

Angle convention

• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 177. LCCN 59014456. Korn and Korn use the (μ, ν, φ) coordinates, but also introduce the degenerate (σ, τ, φ) coordinates.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. p. 182. LCCN 55010911. Like Korn and Korn (1961), but uses colatitude θ = 90° - ν instead of latitude ν.
• Moon PH, Spencer DE (1988). "Oblate spheroidal coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 31–34 (Table 1.07). ISBN 0-387-02732-7. Moon and Spencer use the colatitude convention θ = 90° - ν, and rename φ as ψ.

Unusual convention

• Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. pp. 19–29. ISBN 978-0-7506-2634-7. Treats the oblate spheroidal coordinates as a limiting case of the general ellipsoidal coordinates. Uses (ξ, η, ζ) coordinates that have the units of distance squared.