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: 23.21 Physical Applications

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

4: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(iii) Laplacian

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5: 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

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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

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6: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(iii) Laplacian

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7: 37.19 Other Orthogonal Polynomials of $d$ Variables

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

8: 3.4 Differentiation

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Laplacian

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

9: 18.39 Applications in the Physical Sciences

… ►where ${\nabla }^2$ is the Laplacian (1.5.17). …By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator $T_e$, can be written in spherical coordinates $r,\theta ,\varphi$ as …