spherical coordinates

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1: 14.31 Other Applications

… ►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)). …

2: 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: …

3: 1.5 Calculus of Two or More Variables

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Spherical Coordinates

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1.5.17 $${\nabla }^{2}f=\frac{{\partial }^{2}f}{{\partial x}^2}+\frac{{\partial }^{2}f}{{\partial y}^2}+\frac{{\partial }^{2}f}{{\partial z}^2}=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right).$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\sin z$: sine function, $z$: variable, $\varphi$: longitude, $\rho$: radius and $\theta$: azimuth

Permalink:

http://dlmf.nist.gov/1.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(ii), §1.5(ii), §1.5 and Ch.1

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1.5.41 $$\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}={\rho }^2\sin \theta \mathit{\hspace{1em}}\text{(<span class="ltx\_text hilite">spherical</span><span class="ltx\_text hilite">coordinates</span>)}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\sin z$: sine function, $\varphi$: longitude, $\rho$: radius and $\theta$: azimuth

Permalink:

http://dlmf.nist.gov/1.5.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

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4: 18.39 Physical Applications

… ►when this is solved by separation of variables in spherical coordinates (§1.5(ii)). …

5: 10.73 Physical Applications

… ►The functions ${\mathsf{j}}_n\left(x\right)$, ${\mathsf{y}}_n\left(x\right)$, ${\mathsf{h}}_n^{(1)}\left(x\right)$, and ${\mathsf{h}}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …

6: 31.10 Integral Equations and Representations

… ►A further change of variables, to spherical coordinates, …

7: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►when transformed to sphero-conal coordinates $r,\beta ,\gamma$: … ►

§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: … ►

§29.18(iii) Spherical and Ellipsoidal Harmonics

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8: 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. … ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

9: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). …