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11: 19.33 Triaxial Ellipsoids

… ►

§19.33(i) Surface Area

►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:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

… ►

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:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

…

12: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …

13: 32.16 Physical Applications

… ►Statistical physics, especially classical and quantum spin models, has proved to be a major area for research problems in the modern theory of Painlevé transcendents. …

… ►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis. …

15: 35.9 Applications

… ►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).

16: William P. Reinhardt

… ►This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. …

17: Barry I. Schneider

… ►Schneider’s current research interests span a broad number of areas of theoretical chemistry, atomic and molecular physics, numerical methods and high performance computing. …

18: 1.6 Vectors and Vector-Valued Functions

… ►Area of parallelogram with vectors

\mathbf{a}

and

\mathbf{b}

as sides

=\left\|{\mathbf{a}\times\mathbf{b}}\right\|

. … ►The area of

S

can be found from (1.6.44) by taking

\mathbf{F}(x,y)=-y\mathbf{i}

x\mathbf{j}

, or

-\frac{1}{2}y\mathbf{i}+\frac{1}{2}x\mathbf{j}

. … ►The area

A(S)

of a parametrized smooth surface is given by …The area is independent of the parametrizations. …

19: 5.19 Mathematical Applications

… ►The volume

V

and surface area

S

of the

n

-dimensional sphere of radius

r

are given by …

20: 18.13 Continued Fractions

… ►The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii). …