partial derivative

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1: 1.5 Calculus of Two or More Variables

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►

1.5.9 $$\frac{\partial }{\partial v}f(x(u,v),y(u,v),z(u,v))=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial v}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E9

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

… ►

1.5.38

ⓘ

Symbols:

$det$: determinant, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E38

Encodings:

pMML, png, TeX

See also:

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

… ►

1.5.40

ⓘ

Symbols:

$det$: determinant, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E40

Encodings:

pMML, png, TeX

See also:

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

…

2: 36.10 Differential Equations

… ►

36.10.6 $$\frac{{\partial }^{ln}{\mathrm{\Psi }}_K}{{\partial x_m}^{ln}}={\mathrm{i}}^{n(l-m)}\frac{{\partial }^{mn}{\mathrm{\Psi }}_K}{{\partial x_l}^{mn}},$$ $1\le m\le K$, $1\le l\le K$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $l$: integer, $n$: integer, $m$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(ii)

Permalink:

http://dlmf.nist.gov/36.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10 and Ch.36

… ►

36.10.7 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_2}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_2}{{\partial y}^n}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

… ►

36.10.8 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial y}^n},$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

►

36.10.9 $$\frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial x}^{3n}}={(-1)}^n\frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial z}^n},$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

►

36.10.10 $$\frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial y}^{3n}}={\mathrm{i}}^n\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial z}^{2n}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/36.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

…

3: 19.18 Derivatives and Differential Equations

… ►Let ${\partial }_j=\partial /\partial z_j$, and ${𝐞}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … ►

19.18.6 $$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.18.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.14 $$\frac{{\partial }^{2}w}{{\partial x}^2}=\frac{{\partial }^{2}w}{{\partial y}^2}+\frac{1}{y}\frac{\partial w}{\partial y}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/19.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.15 $$\frac{{\partial }^{2}W}{{\partial t}^2}=\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $W$: wave function

Permalink:

http://dlmf.nist.gov/19.18.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.16 $$\frac{{\partial }^{2}u}{{\partial x}^2}+\frac{{\partial }^{2}u}{{\partial y}^2}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/19.18.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

…

4: 10.38 Derivatives with Respect to Order

… ►

10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). … ►

${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

►

${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=0.$

… ►

10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

…

5: 16.14 Partial Differential Equations

… ►

$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

… ►

$$x(1-x)\frac{{\partial }^2{F}_2}{{\partial x}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_2}{\partial x}-\beta y\frac{\partial F_2}{\partial y}-\alpha \beta F_2=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_2}{{\partial y}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left({\gamma }^{\prime }-(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_2}{\partial y}-{\beta }^{\prime }x\frac{\partial F_2}{\partial x}-\alpha {\beta }^{\prime }{F}_2=0,$$

… ►

$$x(1-x)\frac{{\partial }^2{F}_4}{{\partial x}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-y^2\frac{{\partial }^2{F}_4}{{\partial y}^2}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_4}{\partial x}-(\alpha +\beta +1)y\frac{\partial F_4}{\partial y}-\alpha \beta F_4=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_4}{{\partial y}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-x^2\frac{{\partial }^2{F}_4}{{\partial x}^2}+\left({\gamma }^{\prime }-(\alpha +\beta +1)y\right)\frac{\partial F_4}{\partial y}-(\alpha +\beta +1)x\frac{\partial F_4}{\partial x}-\alpha \beta F_4=0.$$

…

6: 10.15 Derivatives with Respect to Order

… ►

10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

… ►

10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►For $\partial J_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.2.4) and (10.15.3). … ►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

…

7: 12.17 Physical Applications

… ►

12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

ⓘ

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 and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►

12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

…

8: 10.73 Physical Applications

… ►

10.73.1 $${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{{\partial \varphi }^2}+\frac{{\partial }^{2}V}{{\partial z}^2}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.3 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

ⓘ

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 $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

…

9: 20.13 Physical Applications

… ►

20.13.1 $$\partial \theta (z|\tau )/\partial \tau =\kappa {\partial }^2\theta (z|\tau )/{\partial z}^2,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\tau$: lattice parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►

20.13.2 $$\partial \theta /\partial t=\alpha {\partial }^2\theta /{\partial z}^2,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: time, $\alpha$: real parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

…

10: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►

30.13.3 $$h_{\xi }^2={\left(\frac{\partial x}{\partial \xi }\right)}^2+{\left(\frac{\partial y}{\partial \xi }\right)}^2+{\left(\frac{\partial z}{\partial \xi }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{{\xi }^2-1},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

A&S Ref:

21.4.2

Permalink:

http://dlmf.nist.gov/30.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

►

30.13.4 $$h_{\eta }^2={\left(\frac{\partial x}{\partial \eta }\right)}^2+{\left(\frac{\partial y}{\partial \eta }\right)}^2+{\left(\frac{\partial z}{\partial \eta }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{1-{\eta }^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

►

30.13.5 $$h_{\varphi }^2={\left(\frac{\partial x}{\partial \varphi }\right)}^2+{\left(\frac{\partial y}{\partial \varphi }\right)}^2+{\left(\frac{\partial z}{\partial \varphi }\right)}^2=c^2({\xi }^2-1)(1-{\eta }^2).$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\varphi$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

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30.13.6 $${\nabla }^2=\frac{1}{h_{\xi }{h}_{\eta }{h}_{\varphi }}\left(\frac{\partial }{\partial \xi }\left(\frac{h_{\eta }{h}_{\varphi }}{h_{\xi }}\frac{\partial }{\partial \xi }\right)+\frac{\partial }{\partial \eta }\left(\frac{h_{\xi }{h}_{\varphi }}{h_{\eta }}\frac{\partial }{\partial \eta }\right)+\frac{\partial }{\partial \varphi }\left(\frac{h_{\xi }{h}_{\eta }}{h_{\varphi }}\frac{\partial }{\partial \varphi }\right)\right)=\frac{1}{c^2({\xi }^2-{\eta }^2)}\left(\frac{\partial }{\partial \xi }\left(({\xi }^2-1)\frac{\partial }{\partial \xi }\right)+\frac{\partial }{\partial \eta }\left((1-{\eta }^2)\frac{\partial }{\partial \eta }\right)+\frac{{\xi }^2-{\eta }^2}{({\xi }^2-1)(1-{\eta }^2)}\frac{{\partial }^2}{{\partial \varphi }^2}\right).$$

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Symbols:

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

A&S Ref:

21.5.1 (in corrected form)

Permalink:

http://dlmf.nist.gov/30.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iii), §30.13 and Ch.30

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