toroidal coordinates

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21: Bibliography V

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

BibTeX

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22: 22.19 Physical Applications

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22.19.4 $$\frac{d^{2}x(t)}{{dt}^2}=-\frac{dV(x)}{dx},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $V(x)$: potential energy, $x(t)$: coordinate and $t$: time

Permalink:

http://dlmf.nist.gov/22.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.19(ii), §22.19 and Ch.22

►where $V(x)$ is the potential energy, and $x(t)$ is the coordinate as a function of time $t$. … ►

22.19.5 $$V(x)=\pm \frac{1}{2}{x}^2\pm \frac{1}{4}\beta x^4$$

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Symbols:

$V(x)$: potential energy, $x(t)$: coordinate and $\beta$: real positive

Permalink:

http://dlmf.nist.gov/22.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.19(ii), §22.19 and Ch.22

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22.19.6 $$x(t)=a\mathrm{cn}(t\sqrt{1+2\eta },k).$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equation (22.19.6), Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.9):

Originally the term $\sqrt{1+2\eta }$ was given incorrectly as $\sqrt{1+\eta }$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{2+{\eta }^{-1}}$ was given explicitly in the equation.

Reported 2014-04-28 by Svante Janson

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.8 $$x(t)=a\mathrm{dn}(t\sqrt{\eta },k).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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23: 30.2 Differential Equations

… ►In applications involving prolate spheroidal coordinates ${\gamma }^2$ is positive, in applications involving oblate spheroidal coordinates ${\gamma }^2$ is negative; see §§30.13, 30.14. …

24: 23.20 Mathematical Applications

… ►or equivalently, on replacing $x$ by $x/z$ and $y$ by $y/z$ (projective coordinates), into the form ►

23.20.2 $$C:y^{2}z=x^3+ax{z}^2+b{z}^3,$$

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Symbols:

$x$: real part of $z$ , $y$: imaginary part of $z$ , $z$: complex, $C$: elliptic curve, $a$: constant and $b$: constant

Permalink:

http://dlmf.nist.gov/23.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.20(ii), §23.20 and Ch.23

… ►Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …

25: Bibliography T

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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.1, Notations E, Notations E.

Encodings:

BibTeX

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26: 10.73 Physical Applications

… ►In cylindrical coordinates $r$, $\varphi$, $z$, (§1.5(ii) we have … ►On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …

27: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …In elliptical coordinates (28.32.2) becomes (28.32.3). …

28: 26.2 Basic Definitions

… ►A k-dimensional lattice path is a directed path composed of segments that connect vertices in ${\{0,1,2,\mathrm{\dots }\}}^k$ so that each segment increases one coordinate by exactly one unit. …

29: 14.30 Spherical and Spheroidal Harmonics

… ►As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): … ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …

30: 36.6 Scaling Relations

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Indices for $k$-Scaling of Coordinates $x_m$

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