Laplacian

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1: 37.11 Spherical Harmonics

… ►In the case of dimension $d=3$ see §14.30 for spherical harmonics and (1.5.17) for the Laplacian. …The Laplace operator or Laplacian $\Delta ={\nabla }^2$ acting on a smooth function $f(𝐱)=f(x_1,\mathrm{\dots },x_d)$ is given by …The function $f$ is called harmonic if $\Delta f=0$. … ►

§37.11(ii) Spherical Part of Laplacian

… ►The spherical part of the Laplacian $\Delta$ on ${ℝ}^d$ is a differential operator ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_{0,d}$ on ${𝕊}^{d-1}$ defined by …

2: 12.17 Physical Applications

§12.17 Physical Applications

… ►where $k$ is a constant, and ${\nabla }^2$ is the Laplacian …

3: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►The Dunkl Laplacian ${\mathrm{\Delta }}_{\kappa }$ is defined by ►

37.19.2 $${\mathrm{\Delta }}_{\kappa }=T_1^2+\mathrm{\cdots }+T_d^2.$$

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Symbols:

$d$: positive integer, usually $\ge 2$ , ${\mathrm{\Delta }}_{\kappa }$: Dunkl Laplacian and $T_j$: Dunkl operator

Sources:

Dunkl (1989, Definition 1.1, Theorem 1.10); Dunkl and Xu (2014, p. 191)

Permalink:

http://dlmf.nist.gov/37.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19 and Ch.37

… ►

37.19.7 $$⟨f,g⟩={\int }_{{𝔹}^d}{\mathrm{\nabla }}^{s}f(𝐱){\mathrm{\nabla }}^{s}g(𝐱)d𝐱+\sum _{k=0}^{\lfloor \frac{s}{2}\rfloor -1}{\int }_{{𝕊}^{d-1}}{\mathrm{\Delta }}^{k}f(𝝃){\mathrm{\Delta }}^{k}g(𝝃)d\sigma (𝝃),$$

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Defines:

$⟨f,g⟩$: Sobolev inner product on the ball (locally), $𝝃$: $d$-dimensional vector restricted to ${𝕊}^{d-1}$ (locally), $\mathrm{\nabla }$: gradient (locally) and $\mathrm{\Delta }$: Laplacian (locally)

Symbols:

$dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${𝔹}^d$: unit ball, $f$: $d$-variable integrable function on ${𝔹}^d$ and $g$: $d$-variable integrable function on ${𝔹}^d$

Permalink:

http://dlmf.nist.gov/37.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(iv), §37.19 and Ch.37

…

4: 23.21 Physical Applications

… ►The Laplacian operator ${\nabla }^2$ (§1.5(ii)) is given by …

5: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►

§30.14(iii) Laplacian

…

6: 1.5 Calculus of Two or More Variables

… ►The Laplacian is given by ►

1.5.13 $${\nabla }^{2}f=\frac{{\partial }^{2}f}{{\partial x}^2}+\frac{{\partial }^{2}f}{{\partial y}^2}=\frac{{\partial }^{2}f}{{\partial r}^2}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}f}{{\partial \varphi }^2}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: longitude and $r$: radius

Permalink:

http://dlmf.nist.gov/1.5.E13

Encodings:

pMML, png, TeX

See also:

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

… ►

1.5.15 $${\nabla }^{2}f=\frac{{\partial }^{2}f}{{\partial x}^2}+\frac{{\partial }^{2}f}{{\partial y}^2}+\frac{{\partial }^{2}f}{{\partial z}^2}=\frac{{\partial }^{2}f}{{\partial r}^2}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}f}{{\partial \varphi }^2}+\frac{{\partial }^{2}f}{{\partial z}^2}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $\varphi$: longitude and $r$: radius

Permalink:

http://dlmf.nist.gov/1.5.E15

Encodings:

pMML, png, TeX

See also:

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

… ►The Laplacian is given by ►

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 of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $z$: variable, $\varphi$: longitude, $\rho$: radius and $\theta$: azimuth

Referenced by:

§18.39(ii)

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

…

7: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►

§30.13(iii) Laplacian

…

8: 3.4 Differentiation

… ►

Laplacian

… ►Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …

9: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►

37.12.10 $$[t(1-t)D_{tt}+(d-1-(d+\gamma )t)D_t+t^{-1}{\mathrm{\Delta }}_0]u=-n(n+\gamma +d-1)u,$$ $\forall u\in {𝒱}_n({𝕍}_{0,\mathrm{b}}^{d+1},w_{-1,\gamma })$,

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Defines:

$u$: OP on ${𝒱}_n({𝕍}_{0,\mathrm{b}}^{d+1},w)$ (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, ${\mathrm{\Delta }}_0$: spherical part of the Laplacian on Euclidean space, ${𝒱}_n({𝕍}_{0,\mathrm{b}}^{d+1},w_{\beta ,\gamma })$: vector space of orthogonal polynomials on the quadratic surface corresponding to the bounded cone, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕍}_0^{d+1}$: quadratic surface, $t$: real variable in $[a,b]$ , $\gamma$: real parameter and $w_{\beta ,\gamma }$: weight function associated with Jacobi weight

Proved:

Xu (2020, Theorem 7.2)(proved)

Permalink:

http://dlmf.nist.gov/37.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(ii), §37.12(ii), §37.12 and Ch.37

►where $𝐱=t𝝃$ and ${\mathrm{\Delta }}_0$ is the Laplace–Beltrami operator for $𝝃\in {𝕊}^{d-1}$. … ►

37.12.15 $$[t{D}_{tt}+(d-1-t)D_t+t^{-1}{\mathrm{\Delta }}_0]u=-nu,$$ $\forall u\in {𝒱}_n({𝕍}_{0,\mathrm{u}}^{d+1},w_{-1})$,

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Defines:

$u$: OP on ${𝒱}_n({𝕍}_{0,\mathrm{u}}^{d+1},w_{-1})$ (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, ${\mathrm{\Delta }}_0$: spherical part of the Laplacian on Euclidean space, ${𝒱}_n({𝕍}_{0,\mathrm{u}}^{d+1},w_{\beta })$: vector space of orthogonal polynomials on the quadratic surface corresponding to the unbounded cone, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕍}_0^{d+1}$: quadratic surface, $t$: real variable in $[a,b]$ and $w_{\beta }$: weight function associated with Laguerre weight

Proved:

Xu (2020, Theorem 7.3)(proved)

Permalink:

http://dlmf.nist.gov/37.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(iii), §37.12(iii), §37.12 and Ch.37

►where $𝐱=t𝝃$ and ${\mathrm{\Delta }}_0$ is the Laplace–Beltrami operator for $𝝃\in {𝕊}^{d-1}$. …

10: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►

37.17.10 $$\left(\frac{1}{2}\mathrm{\Delta }-\sum _{\mathrm{\ell }=1}^d{x}_{\mathrm{\ell }}{D}_{x_{\mathrm{\ell }}}\right)u(𝐱)=-nu(𝐱),$$ $u\in {𝒱}_n\left({ℝ}^d\right)$.

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Defines:

$\mathrm{\Delta }$: Laplacian (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $ℝ$: real line, ${𝒱}_n\left({ℝ}^d\right)$: vector space of $d$-variable orthogonal polynomials with respect to the Gaussian weight, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $u$: orthogonal polynomial in $d$-variables with respect to the Gaussian weight and $x$: real variable

Proved:

Dunkl and Xu (2014, (5.1.7))(proved)

Referenced by:

§37.13(ii), §37.17

Permalink:

http://dlmf.nist.gov/37.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(iii), §37.17 and Ch.37

…