partial derivatives

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21: 14.30 Spherical and Spheroidal Harmonics

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14.30.10 $$\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial W}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial W}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}W}{{\partial \varphi }^2}=0,$$

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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 and $W(\rho ,\theta ,\varphi )$: solution

Permalink:

http://dlmf.nist.gov/14.30.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.12 $$L^2=-{\mathrm{\hslash }}^2\left(\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{{\partial \varphi }^2}\right),$$

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Defines:

$L$: angular momentum operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\sin z$: sine function

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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22: 21.9 Integrable Equations

… ►Here, and in what follows, $x,y$, and $t$ suffixes indicate partial derivatives. … ►

21.9.4 $$u(x,y,t)=c+2\frac{{\partial }^2}{{\partial x}^2}\ln \left(\theta \left(𝐤x+𝐥y+𝝎t+\mathit{\varphi }\right|𝛀)\right),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝛀$: a Riemann matrix, $u(x,y,t)$: elevation and $c$: complex constant

Referenced by:

§21.9

Permalink:

http://dlmf.nist.gov/21.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

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23: 7.19 Voigt Functions

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7.19.8 $$𝖵(x,t)=x𝖴(x,t)+2t\frac{\partial 𝖴(x,t)}{\partial x},$$

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(iii), §7.19 and Ch.7

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7.19.9 $$𝖴(x,t)=1-x𝖵(x,t)-2t\frac{\partial 𝖵(x,t)}{\partial x}.$$

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(iii), §7.19 and Ch.7

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24: 28.33 Physical Applications

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28.33.1 $$\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}-\frac{\rho }{\tau }\frac{{\partial }^{2}W}{{\partial t}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $\rho$: mass, $\tau$: radial tension and $W(x,y,t)$: solution

Permalink:

http://dlmf.nist.gov/28.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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25: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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30.14.6 $${\nabla }^2=\frac{1}{c^2({\xi }^2+{\eta }^2)}\left(\frac{\partial }{\partial \xi }\left(({\xi }^2+1)\frac{\partial }{\partial \xi }\right)+\frac{\partial }{\partial \eta }\left((1-{\eta }^2)\frac{\partial }{\partial \eta }\right)+\frac{{\xi }^2+{\eta }^2}{({\xi }^2+1)(1-{\eta }^2)}\frac{{\partial }^2}{{\partial \varphi }^2}\right).$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate, $\varphi$: oblate spheroidal coordinate and $c$: positive constant

A&S Ref:

21.5.2 (in corrected form)

Permalink:

http://dlmf.nist.gov/30.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(iii), §30.14 and Ch.30

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26: 1.9 Calculus of a Complex Variable

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${\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial y}},$

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${\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{\partial v}{\partial x}}$

… ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►

1.9.26 $$\frac{{\partial }^{2}u}{{\partial x}^2}+\frac{{\partial }^{2}u}{{\partial y}^2}=\frac{{\partial }^{2}v}{{\partial x}^2}+\frac{{\partial }^{2}v}{{\partial y}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $u(x,y)$: function and $v(x,y)$: function

A&S Ref:

3.7.32

Permalink:

http://dlmf.nist.gov/1.9.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

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1.9.27 $$\frac{{\partial }^{2}u}{{\partial r}^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{{\partial \theta }^2}=0$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $r$: radius, $\theta$: angle and $u(x,y)$: function

A&S Ref:

3.7.33

Referenced by:

§1.15(iii)

Permalink:

http://dlmf.nist.gov/1.9.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

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27: 23.15 Definitions

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23.15.8 $${\theta }_1^{\prime }(0,q)={\partial {\theta }_1(z,q)/\partial z|}_{z=0}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q$: nome and $z$: complex

Permalink:

http://dlmf.nist.gov/23.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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28: 21.7 Riemann Surfaces

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21.7.7 $$({\frac{\partial }{\partial z_1}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}},\mathrm{\dots },{\frac{\partial }{\partial z_g}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}})\ne \mathrm{𝟎}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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29: 2.4 Contour Integrals

… ►Suppose that on the integration path $𝒫$ there are two simple zeros of $\partial p(\alpha ,t)/\partial t$ that coincide for a certain value $\widehat{\alpha }$ of $\alpha$. … ►with $a$ and $b$ chosen so that the zeros of $\partial p(\alpha ,t)/\partial t$ correspond to the zeros $w_1(\alpha ),w_2(\alpha )$, say, of the quadratic $w^2+2aw+b$. … ►

2.4.20 $$f(\alpha ,w)=q(\alpha ,t)\frac{dt}{dw}=q(\alpha ,t)\frac{w^2+2aw+b}{\partial p(\alpha ,t)/\partial t}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $p(t)$: analytic function, $q(t)$: analytic function, $w$: change of variable, $a$ , $b$ and $f(\alpha ,w)$: function

Permalink:

http://dlmf.nist.gov/2.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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30: 31.9 Orthogonality

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31.9.3 $${\theta }_m={(1-{\mathrm{e}}^{2\pi \mathrm{i}\gamma })(1-{\mathrm{e}}^{2\pi \mathrm{i}\delta }){\zeta }^{\gamma }{(1-\zeta )}^{\delta }{(\zeta -a)}^{ϵ}\frac{f_0(q,\zeta )}{f_1(q,\zeta )}\frac{\partial }{\partial q}𝒲\left\{f_0(q,\zeta ),f_1(q,\zeta )\right\}|}_{q=q_m},$$

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Defines:

${\theta }_m$: normalization constant (locally)

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\zeta$: point and $f_0(q_m,z)$, $f_1(q_m,z)$: coefficients

Permalink:

http://dlmf.nist.gov/31.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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