# n-dimensional sphere

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## 1—10 of 12 matching pages

##### 1: 5.19 Mathematical Applications
###### §5.19(iii) $n$-DimensionalSphere
The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by …
##### 2: 1.1 Special Notation
 $x,y$ real variables. … the space of all $n$-dimensional vectors. …
##### 3: Bibliography R
• M. Robnik (1980) An extremum property of the $n$-dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.
• ##### 4: 1.2 Elementary Algebra
Let $\left\|{\mathbf{x}}\right\|=\left\|{\mathbf{x}}\right\|_{2}$ the $l^{2}$ norm, and $\mathbf{E}_{n}$ the space of all $n$-dimensional vectors. …
1.2.67 $\left\|{\mathbf{A}}\right\|=\max_{\mathbf{x}\in\mathbf{E}_{n}\setminus\left\{% \boldsymbol{{0}}\right\}}\frac{\left\|{\mathbf{A}\mathbf{x}}\right\|}{\left\|{% \mathbf{x}}\right\|}=\max_{\left\|{\mathbf{x}}\right\|=1}\left\|{\mathbf{A}% \mathbf{x}}\right\|.$
##### 5: 1.3 Determinants, Linear Operators, and Spectral Expansions
Square matices can be seen as linear operators because $\mathbf{A}(\alpha\mathbf{a}+\beta\mathbf{b})=\alpha\mathbf{A}\mathbf{a}+\beta% \mathbf{A}\mathbf{b}$ for all $\alpha,\beta\in\mathbb{C}$ and $\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}$, the space of all $n$-dimensional vectors. …
##### 6: 14.31 Other Applications
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ 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)). …
##### 7: 31.17 Physical Applications
Consider the following spectral problem on the sphere $S_{2}$: $\mathbf{x}^{2}=x_{s}^{2}+x_{t}^{2}+x_{u}^{2}=R^{2}$. …
##### 8: 19.37 Tables
Here $\sigma^{2}=\tfrac{2}{3}((\ln a)^{2}+(\ln b)^{2}+(\ln c)^{2})$, $\cos\left(3\gamma\right)=(4/\sigma^{3})(\ln a)(\ln b)(\ln c)$, and $a,b,c$ are semiaxes of an ellipsoid with the same volume as the unit sphere. …
##### 9: Bibliography N
• H. M. Nussenzveig (1965) High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.
• ##### 10: 1.6 Vectors and Vector-Valued Functions
For a sphere $x=\rho\sin\theta\cos\phi$, $y=\rho\sin\theta\sin\phi$, $z=\rho\cos\theta$,
1.6.50 $\left\|{\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}}\right\|=\rho^{2}\left|\sin% \theta\right|.$