partial%20differentiation

SearchAdvancedHelp

(0.001 seconds)

21—30 of 233 matching pages

21: 1.6 Vectors and Vector-Valued Functions

… ►

1.6.19 $$\nabla =𝐢\frac{\partial }{\partial x}+𝐣\frac{\partial }{\partial y}+𝐤\frac{\partial }{\partial z}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

1.6.20 $$\mathrm{grad}f=\nabla f=\frac{\partial f}{\partial x}𝐢+\frac{\partial f}{\partial y}𝐣+\frac{\partial f}{\partial z}𝐤.$$

ⓘ

Defines:

$\mathrm{grad}$: gradient of differentiable scalar function

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

1.6.22

ⓘ

Defines:

$\mathrm{curl}$: of vector-valued function

Symbols:

$det$: determinant, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector, $𝐤$: unit vector and $F_j$: vector function components

Permalink:

http://dlmf.nist.gov/1.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … ►where $\partial g/\partial n=\nabla g\cdot 𝐧$ is the derivative of $g$ normal to the surface outwards from $V$ and $𝐧$ is the unit outer normal vector. …

22: 5.19 Mathematical Applications

… ►By decomposition into partial fractions (§1.2(iii)) …

23: 1.10 Functions of a Complex Variable

… ►Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup \partial D$. … ► … ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. … ►

§1.10(x) Infinite Partial Fractions

… ►

Mittag-Leffler’s Expansion

…

24: 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

… ►

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

ⓘ

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

…

25: 31.10 Integral Equations and Representations

… ►and the kernel $𝒦(z,t)$ is a solution of the partial differential equation … ►

31.10.4 $${𝒟}_z=z(z-1)(z-a)({\partial }^2/{\partial z}^2)+\left(\gamma (z-1)(z-a)+\delta z(z-a)+ϵz(z-1)\right)(\partial /\partial z)+\alpha \beta z.$$

ⓘ

Defines:

${𝒟}_z$: Heun’s operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.10(ii)

Permalink:

http://dlmf.nist.gov/31.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

… ►

31.10.8 $${\sin }^2\theta \left(\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\left((1-2\gamma )\tan \theta +2(\delta +ϵ-\frac{1}{2})\cot \theta \right)\frac{\partial 𝒦}{\partial \theta }-4\alpha \beta 𝒦\right)+\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\left((1-2\delta )\cot \varphi -(1-2ϵ)\tan \varphi \right)\frac{\partial 𝒦}{\partial \varphi }=0.$$

ⓘ

Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle and $\varphi$: angle

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

… ►and the kernel $𝒦(z;s,t)$ is a solution of the partial differential equation … ►

31.10.18 $$\frac{{\partial }^2𝒦}{{\partial u}^2}+\frac{{\partial }^2𝒦}{{\partial v}^2}+\frac{{\partial }^2𝒦}{{\partial w}^2}+\frac{2\gamma -1}{u}\frac{\partial 𝒦}{\partial u}+\frac{2\delta -1}{v}\frac{\partial 𝒦}{\partial v}+\frac{2ϵ-1}{w}\frac{\partial 𝒦}{\partial w}=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: 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, $𝒦(z;s,t)$: kernel, $u$ , $v$ and $w$

Permalink:

http://dlmf.nist.gov/31.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

…

26: 14.11 Derivatives with Respect to Degree or Order

… ►

14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►

14.11.2 $$\frac{\partial }{\partial \nu }{𝖰}_{\nu }^{\mu }\left(x\right)=-\frac{1}{2}{\pi }^2{𝖯}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{𝖰}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(x)+\frac{1}{2}\csc \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(-x),$$

ⓘ

Symbols:

: Ferrers function of the first kind, : Ferrers 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 $x$ , $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

… ►

14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►

14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

…

27: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►

10.40.8 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}\frac{\nu {\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k(\nu )}{{(8z)}^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: 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 $x$ , $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_k(\nu )$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

…

28: 36.12 Uniform Approximation of Integrals

… ►Also, $f$ is real analytic, and ${\partial }^{K+2}f/{\partial u}^{K+2}>0$ for all $𝐲$ such that all $K+1$ critical points coincide. If , then we may evaluate the complex conjugate of $I$ for real values of $𝐲$ and $g$, and obtain $I$ by conjugation and analytic continuation. … ►

36.12.2 $$\frac{\partial }{\partial u}f(u_j(𝐲);𝐲)=0.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/36.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.10 $$G_n(𝐲)=g(t_n(𝐲),𝐲)\sqrt{\frac{{\partial }^2{\mathrm{\Phi }}_K(t_n(𝐱(𝐲));𝐱(𝐲))/{\partial t}^2}{{\partial }^{2}f(u_n(𝐲))/{\partial u}^2}}.$$

ⓘ

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $m$: integer, $t$: variable, $K$: codimension, $g(u,𝐲)$: function, $f(u,𝐲)$: function, $u(t;𝐲)$: mapping, $𝐱(𝐲)$: function and $t_j(𝐱)$: solutions

Referenced by:

§36.12(i)

Permalink:

http://dlmf.nist.gov/36.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►

$f_{\pm }^{\prime \prime }={\displaystyle \frac{{\partial }^2}{{\partial u}^2}}f(u_{\pm }(y),y),$

…

29: 25.11 Hurwitz Zeta Function

… ►

►►

Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

… ►

25.11.17 $$\frac{\partial }{\partial a}\zeta (s,a)=-s\zeta (s+1,a),$$ $s\ne 0,1$; $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

derivative

Proof sketch:

Derivable by differentiating (25.11.1) with respect to $a$.

Referenced by:

(25.11.10)

Permalink:

http://dlmf.nist.gov/25.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

… ►

25.11.24 $$\sum _{r=1}^{k-1}{\zeta }^{\prime }(s,\frac{r}{k})=(k^s-1){\zeta }^{\prime }\left(s\right)+k^s\zeta \left(s\right)\ln k,$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Proof sketch:

Derivable from (25.11.15) with $a=1/k$, multiplying by $k^s$, differentiating with respect to $s$, and using (25.11.2).

Referenced by:

(25.11.21), §25.11(vi)

Permalink:

http://dlmf.nist.gov/25.11.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

…

30: 20 Theta Functions

Chapter 20 Theta Functions

…