19 Elliptic Integrals Applications 19.32 Conformal Map onto a Rectangle 19.34 Mutual Inductance of Coaxial Circles

§19.33 Triaxial Ellipsoids

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Keywords:: ellipsoid, triaxial
Referenced by:: §19.15
Permalink:: http://dlmf.nist.gov/19.33
See also:: Annotations for Ch.19

§19.33(i) Surface Area
§19.33(ii) Potential of a Charged Conducting Ellipsoid
§19.33(iii) Depolarization Factors
§19.33(iv) Self-Energy of an Ellipsoidal Distribution

§19.33(i) Surface Area

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Keywords:: ellipsoid, surface area
Notes:: For a short proof see Carlson (1977b, p. 271). For other proofs see Watson (1935b), Bowman (1953, pp. 31–32), and Carlson (1964, p. 417).
Permalink:: http://dlmf.nist.gov/19.33.i
See also:: Annotations for §19.33 and Ch.19

The surface area of an ellipsoid with semiaxes $a, b, c$ , and volume $V=4\pi abc/3$ is given by

19.33.1		$S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right),$
ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $V$ : volume and $S$ : area Referenced by: §19.33(i) Permalink: http://dlmf.nist.gov/19.33.E1 Encodings: TeX, pMML, png See also: Annotations for §19.33(i), §19.33 and Ch.19

or equivalently,

19.33.2		$\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E\left(\phi,k\right){\sin}^{2}% \phi+F\left(\phi,k\right){\cos}^{2}\phi\right),$
$a\geq b\geq c$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $S$ : area Permalink: http://dlmf.nist.gov/19.33.E2 Encodings: TeX, pMML, png See also: Annotations for §19.33(i), §19.33 and Ch.19

where

19.33.3	$\displaystyle\cos\phi$	$\displaystyle=\frac{c}{a},$
19.33.3	$\displaystyle k^{2}$	$\displaystyle=\frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\phi$ : real or complex argument and $k$ : real or complex modulus Permalink: http://dlmf.nist.gov/19.33.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.33(i), §19.33 and Ch.19

Application of (19.16.23) transforms the last quantity in (19.30.5) into a two-dimensional analog of (19.33.1).

For additional geometrical properties of ellipsoids (and ellipses), see Carlson (1964, p. 417).

§19.33(ii) Potential of a Charged Conducting Ellipsoid

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Keywords:: applications, capacity, ellipsoid, physical, potential, potential theory, symmetric elliptic integrals
Notes:: See Carlson (1977b, (9.4-10)).
Permalink:: http://dlmf.nist.gov/19.33.ii
See also:: Annotations for §19.33 and Ch.19

If a conducting ellipsoid with semiaxes $a, b, c$ bears an electric charge $Q$ , then the equipotential surfaces in the exterior region are confocal ellipsoids:

19.33.4		$\frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+% \lambda}=1,$
$\lambda\geq 0$ .
ⓘ Permalink: http://dlmf.nist.gov/19.33.E4 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

The potential is

19.33.5		$V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $Q$ : charge and $V(\lambda)$ : potential Permalink: http://dlmf.nist.gov/19.33.E5 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

and the electric capacity $C=Q/V(0)$ is given by

19.33.6		$1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $C$ : electric capacity Permalink: http://dlmf.nist.gov/19.33.E6 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

A conducting elliptic disk is included as the case $c=0$ .

§19.33(iii) Depolarization Factors

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Keywords:: depolarization, ellipsoid
Notes:: For the first equality in (19.33.7) see Becker and Sauter (1964, p. 106).
Permalink:: http://dlmf.nist.gov/19.33.iii
See also:: Annotations for §19.33 and Ch.19

Let a homogeneous magnetic ellipsoid with semiaxes $a, b, c$ , volume $V=4\pi abc/3$ , and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$ . The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_{c}\chi)$ , where $L_{c}$ is the demagnetizing factor, given in cgs units by

19.33.7		$L_{c}=2\pi abc\int_{0}^{\infty}\frac{\,\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda% )(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b^{2},c^{2}\right).$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L_{a}$ , $L_{b}$ : factor and $V$ : volume Referenced by: §19.15, §19.33(iii) Permalink: http://dlmf.nist.gov/19.33.E7 Encodings: TeX, pMML, png See also: Annotations for §19.33(iii), §19.33 and Ch.19

The same result holds for a homogeneous dielectric ellipsoid in an electric field. By (19.21.8),

19.33.8		$L_{a}+L_{b}+L_{c}=4\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter and $L_{a}$ , $L_{b}$ : factor Permalink: http://dlmf.nist.gov/19.33.E8 Encodings: TeX, pMML, png See also: Annotations for §19.33(iii), §19.33 and Ch.19

where $L_{a}$ and $L_{b}$ are obtained from $L_{c}$ by permutation of $a$ , $b$ , and $c$ . Expressions in terms of Legendre’s integrals, numerical tables, and further references are given by Cronemeyer (1991).

§19.33(iv) Self-Energy of an Ellipsoidal Distribution

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Keywords:: ellipsoid, self-energy
Notes:: See Carlson (1961a) and Carlson (1977b, Ex. 9.4-3 and p. 313).
Permalink:: http://dlmf.nist.gov/19.33.iv
See also:: Annotations for §19.33 and Ch.19

Ellipsoidal distributions of charge or mass are used to model certain atomic nuclei and some elliptical galaxies. Let the density of charge or mass be

19.33.9		$\rho(x,y,z)=f\left(\sqrt{(x^{2}/\alpha^{2})+(y^{2}/\beta^{2})+(z^{2}/\gamma^{2% })}\right),$
ⓘ Symbols: $\rho(x,y,z)$ : density, $f(t)$ : density, $\alpha$ : positive constant, $\beta$ : positive constant and $\gamma$ : positive constant Permalink: http://dlmf.nist.gov/19.33.E9 Encodings: TeX, pMML, png See also: Annotations for §19.33(iv), §19.33 and Ch.19

where $\alpha,\beta,\gamma$ are dimensionless positive constants. The contours of constant density are a family of similar, rather than confocal, ellipsoids. In suitable units the self-energy of the distribution is given by

19.33.10		$U=\frac{1}{2}\int_{{\mathbb{R}}^{6}}\frac{\rho(x,y,z)\rho(x^{\prime},y^{\prime% },z^{\prime})\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}x^{\prime}\,% \mathrm{d}y^{\prime}\,\mathrm{d}z^{\prime}}{\sqrt{(x-x^{\prime})^{2}+(y-y^{% \prime})^{2}+(z-z^{\prime})^{2}}}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{R}$ : real line, $\rho(x,y,z)$ : density and $U$ : energy Permalink: http://dlmf.nist.gov/19.33.E10 Encodings: TeX, pMML, png See also: Annotations for §19.33(iv), §19.33 and Ch.19

Subject to mild conditions on $f$ this becomes

19.33.11		$U=\tfrac{1}{2}(\alpha\beta\gamma)^{2}R_{F}\left(\alpha^{2},\beta^{2},\gamma^{2% }\right)\int_{0}^{\infty}(g(r))^{2}\,\mathrm{d}r,$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\alpha$ : positive constant, $\beta$ : positive constant, $\gamma$ : positive constant, $U$ : energy and $g(r)$ : function Permalink: http://dlmf.nist.gov/19.33.E11 Encodings: TeX, pMML, png See also: Annotations for §19.33(iv), §19.33 and Ch.19

where

19.33.12		$g(r)=4\pi\int_{r}^{\infty}f(t)t\,\mathrm{d}t.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $f(t)$ : density and $g(r)$ : function Permalink: http://dlmf.nist.gov/19.33.E12 Encodings: TeX, pMML, png See also: Annotations for §19.33(iv), §19.33 and Ch.19

19.32 Conformal Map onto a Rectangle 19.34 Mutual Inductance of Coaxial Circles

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§19.33 Triaxial Ellipsoids

Contents

§19.33(i) Surface Area

§19.33(ii) Potential of a Charged Conducting Ellipsoid

§19.33(iii) Depolarization Factors

§19.33(iv) Self-Energy of an Ellipsoidal Distribution