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

§19.33 Triaxial Ellipsoids

Keywords:: ellipsoid
Referenced by:: §19.15
Permalink:: http://dlmf.nist.gov/19.33

§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

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).
Keywords:: ellipsoid
Permalink:: http://dlmf.nist.gov/19.33.i

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

19.33.1 $S=3V\mathop{R_{G}\/}\nolimits\!\left(a^{{-2}},b^{{-2}},c^{{-2}}\right),$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, : volume and : area
Referenced by:: §19.33(i)
Permalink:: http://dlmf.nist.gov/19.33.E1
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or equivalently,

19.33.2 $\frac{S}{2\pi}=c^{2}+\frac{ab}{\mathop{\sin\/}\nolimits\phi}\left(\mathop{E\/}% \nolimits\!\left(\phi,k\right){\mathop{\sin\/}\nolimits^{{2}}}\phi+\mathop{F\/% }\nolimits\!\left(\phi,k\right){\mathop{\cos\/}\nolimits^{{2}}}\phi\right),$ $a\geq b\geq c$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and : area
Permalink:: http://dlmf.nist.gov/19.33.E2
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where

19.33.3

$\mathop{\cos\/}\nolimits\phi=\frac{c}{a},$

$k^{2}=\frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.33.E3
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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

Notes:: See Carlson (1977b, (9.4-10)).
Keywords:: ellipsoid, potential theory, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.33.ii

If a conducting ellipsoid with semiaxes bears an electric charge , 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
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The potential is

19.33.5 $V(\lambda)=Q\mathop{R_{F}\/}\nolimits\!\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}% +\lambda\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : charge and $V(\lambda)$ : potential
Permalink:: http://dlmf.nist.gov/19.33.E5
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and the electric capacity is given by

19.33.6 $1/C=\mathop{R_{F}\/}\nolimits\!\left(a^{2},b^{2},c^{2}\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and : electric capacity
Permalink:: http://dlmf.nist.gov/19.33.E6
Encodings:: TeX, pMML, png

A conducting elliptic disk is included as the case .

§19.33(iii) Depolarization Factors

Notes:: For the first equality in (19.33.7) see Becker and Sauter (1964, p. 106).
Keywords:: ellipsoid
Permalink:: http://dlmf.nist.gov/19.33.iii

Let a homogeneous magnetic ellipsoid with semiaxes , volume $V=4\pi abc/3$ , and susceptibility $\chi$ be placed in a previously uniform magnetic field parallel to the principal axis with semiaxis . 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 $L_{c}=2\pi abc\int_{0}^{{\infty}}\frac{d\lambda}{\sqrt{(a^{2}+\lambda)(b^{2}+% \lambda)(c^{2}+\lambda)^{3}}}=V\mathop{R_{D}\/}\nolimits\!\left(a^{2},b^{2},c^% {2}\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, : differential of , $\int$ : integral, $L_{a}$ , $L_{b}$ : factor and : volume
Referenced by:: §19.15, §19.33(iii)
Permalink:: http://dlmf.nist.gov/19.33.E7
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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:: $L_{a}$ , $L_{b}$ : factor
Permalink:: http://dlmf.nist.gov/19.33.E8
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where $L_{a}$ and $L_{b}$ are obtained from $L_{c}$ by permutation of , , and . 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

Notes:: See Carlson (1961a) and Carlson (1977b, Ex. 9.4-3 and p. 313).
Keywords:: ellipsoid
Permalink:: http://dlmf.nist.gov/19.33.iv

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, : density, $\alpha$ : positive constant, $\beta$ : positive constant and $\gamma$ : positive constant
Permalink:: http://dlmf.nist.gov/19.33.E9
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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 $U=\frac{1}{2}\int_{{\Real^{6}}}\frac{\rho(x,y,z)\rho(x^{{\prime}},y^{{\prime}}% ,z^{{\prime}})dxdydzdx^{{\prime}}dy^{{\prime}}dz^{{\prime}}}{\sqrt{(x-x^{{% \prime}})^{2}+(y-y^{{\prime}})^{2}+(z-z^{{\prime}})^{2}}}.$

Symbols:: : differential of , $\int$ : integral, $\Real$ : real line (excluding infinity), $\rho(x,y,z)$ : density and : energy
Permalink:: http://dlmf.nist.gov/19.33.E10
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Subject to mild conditions on this becomes

19.33.11 $U=\tfrac{1}{2}(\alpha\beta\gamma)^{2}\mathop{R_{F}\/}\nolimits\!\left(\alpha^{% 2},\beta^{2},\gamma^{2}\right)\int_{0}^{{\infty}}(g(r))^{2}dr,$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\int$ : integral, $\alpha$ : positive constant, $\beta$ : positive constant, $\gamma$ : positive constant, : energy and : function
Permalink:: http://dlmf.nist.gov/19.33.E11
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where

19.33.12 $g(r)=4\pi\int_{r}^{{\infty}}f(t)tdt.$

Symbols:: : differential of , $\int$ : integral, : density and : function
Permalink:: http://dlmf.nist.gov/19.33.E12
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.32 Conformal Map onto a Rectangle 19.34 Mutual Inductance of Coaxial Circles

§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