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1: 5.19 Mathematical Applications

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§5.19(iii) $n$-Dimensional Sphere

►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by …

2: 14.31 Other Applications

… ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

3: 31.17 Physical Applications

… ►Consider the following spectral problem on the sphere $S_2$: ${𝐱}^2=x_s^2+x_t^2+x_u^2=R^2$. …

4: 19.37 Tables

… ►Here ${\sigma }^2=\frac{2}{3}({(\ln a)}^2+{(\ln b)}^2+{(\ln c)}^2)$, $\cos \left(3\gamma \right)=(4/{\sigma }^3)(\ln a)(\ln b)(\ln c)$, and $a,b,c$ are semiaxes of an ellipsoid with the same volume as the unit sphere. …

5: Bibliography N

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High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.

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External Links:

Document, MathReview (E. Ar)

Referenced by:

§14.31(iii).

Encodings:

BibTeX

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6: Bibliography R

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An extremum property of the $n$-dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.

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External Links:

ISSN 0305-4470, Document, MathReview

Referenced by:

§5.19(iii).

Encodings:

BibTeX

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7: 1.6 Vectors and Vector-Valued Functions

… ►For a sphere $x=\rho \sin \theta \cos \varphi$, $y=\rho \sin \theta \sin \varphi$, $z=\rho \cos \theta$, ►

1.6.50 $$\Vert {𝐓}_{\theta }\times {𝐓}_{\varphi }\Vert ={\rho }^2\left|\sin \theta \right|.$$

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Symbols:

$\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\rho$: radius, $\theta$: angle, $\varphi$: angle and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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8: Bibliography D

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Atomic motions in a rigid sphere gas as a problem in neutron transport. Nucl. Sci. Eng. 24 (2), pp. 142–152.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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9: Bibliography K

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Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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