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1: About Color Map

… ►The following figure illustrates the piece-wise linear mapping of the height to each of the color components red, green and blue, written as $⟨R,G,B⟩$. … ►Mathematically, we scale the height to $h$ lying in the interval $[0,4]$ and the components are computed as follows …

2: 16.23 Mathematical Applications

… ►In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. …

3: 1.6 Vectors and Vector-Valued Functions

… ►Much vector algebra involves summation over suffices of products of vector components. … ►The divergence of a differentiable vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ is ►

1.6.21 $$\mathrm{div}𝐅=\nabla \cdot 𝐅=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$$

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Defines:

$\mathrm{div}$: divergence of vector-valued function

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable and $F_j$: vector function components

Permalink:

http://dlmf.nist.gov/1.6.E21

Encodings:

pMML, png, TeX

See also:

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

… ►The line integral of a vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ along $𝐜$ is given by … ►

1.6.43 $$𝐅(x,y)=F_1(x,y)𝐢+F_2(x,y)𝐣$$

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Defines:

$𝐅(x,y)$: vector (locally)

Symbols:

$𝐢$: unit vector, $𝐣$: unit vector and $F_j$: vector function components

Permalink:

http://dlmf.nist.gov/1.6.E43

Encodings:

pMML, png, TeX

See also:

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

…

4: 19.19 Taylor and Related Series

… ►The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. …

5: 21.2 Definitions

… ►

21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

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See also:

Annotations for §21.2(i), §21.2 and Ch.21

… ► $\theta \left(𝐳\right|𝛀)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. …

6: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.11_5 $${\mathrm{L}}_z{Y}_{l,m}=\mathrm{\hslash }m{Y}_{l,m},$$ $m=-l,-1+1,\mathrm{\dots },0,\mathrm{\dots },l-1,l$,

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and ${\mathrm{L}}_z$: $z$ component of the angular momentum operator

Referenced by:

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

Permalink:

http://dlmf.nist.gov/14.30.E11_5

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

… ►and ${\mathrm{L}}_z$ is the $z$ component of the angular momentum operator ►

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

…

7: 19.24 Inequalities

… ►If $a$ ($\ne 0$) is real, all components of $𝐛$ and $𝐳$ are positive, and the components of $z$ are not all equal, then …

8: Bibliography J

… ►

The birth of the giant component. Random Structures Algorithms 4 (3), pp. 231–358.

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Notes:

With an introduction by the editors

External Links:

ISSN 1042-9832, Document, MathReview (Alan M. Frieze), ZentralBlatt

Referenced by:

§16.23(ii), §9.13(ii).

Encodings:

BibTeX

…

9: 21.7 Riemann Surfaces

… ►where $P_1$ and $P_2$ are points on $\mathrm{\Gamma }$, $𝝎=({\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_g)$, and the path of integration on $\mathrm{\Gamma }$ from $P_1$ to $P_2$ is identical for all components. … ►where again all integration paths are identical for all components. …

10: 19.28 Integrals of Elliptic Integrals

… ►To replace a single component of $𝐳$ in $R_{-a}(𝐛;𝐳)$ by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)). …