# spherical coordinates

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## 9 matching pages

##### 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 coordinates1.5(ii)): … Here, in spherical coordinates, $\mathrm{L}^{2}$ is the squared angular momentum operator: …
##### 3: 1.5 Calculus of Two or More Variables
###### SphericalCoordinates
1.5.17 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{% \partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}={\frac{1}{\rho^{2}}% \frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial f}{\partial\rho}% \right)+\frac{1}{\rho^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}f}{{\partial\phi% }^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin% \theta\frac{\partial f}{\partial\theta}\right).$
1.5.41 $\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}=\rho^{2}\sin\theta\quad% \text{(spherical coordinates)}.$
##### 4: 18.39 Physical Applications
when this is solved by separation of variables in spherical coordinates1.5(ii)). …
##### 5: 10.73 Physical Applications
The functions $\mathsf{j}_{n}\left(x\right)$, $\mathsf{y}_{n}\left(x\right)$, ${\mathsf{h}^{(1)}_{n}}\left(x\right)$, and ${\mathsf{h}^{(2)}_{n}}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho,\theta,\phi$1.5(ii)): …
##### 6: 31.10 Integral Equations and Representations
A further change of variables, to spherical coordinates, …
##### 7: 29.18 Mathematical Applications
###### §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 $\alpha,\beta,\gamma$: …
##### 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, $x=r\cos\phi$, $y=r\sin\phi$, the lemniscate is given by $r^{2}=\cos\left(2\phi\right)$, $0\leq\phi\leq 2\pi$. … 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). …