Levi-Civita symbol for vectors

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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

… ►

Levi-Civita Symbol

►

1.6.14

\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even % permutation of }1,2,3,\\ -1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\ \phantom{-}0,&\text{otherwise}.\end{cases}

ⓘ

Defines:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol
Symbols:: $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

1.6.17

\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk\ell}\mathbf{e}_{\ell};

ⓘ

Symbols:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol, $j$ : integer, $k$ : integer and $\mathbf{e}_{j}$ : unit vectors
Permalink:: http://dlmf.nist.gov/1.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

1.6.18

a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\epsilon_{jk\ell}a_{j}b_{k}% \mathbf{e}_{\ell};

ⓘ

Symbols:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol, $j$ : integer, $k$ : integer and $\mathbf{e}_{j}$ : unit vectors
Permalink:: http://dlmf.nist.gov/1.6.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

…

2: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: 34.2 Definition: $\mathit{3j}$ Symbol

§34.2 Definition: $\mathit{3j}$ Symbol

►The quantities

j_{1},j_{2},j_{3}

in the

\mathit{3j}

symbol are called angular momenta. …They therefore satisfy the triangle conditions …The corresponding projective quantum numbers

m_{1},m_{2},m_{3}

are given by … ►When both conditions are satisfied the

\mathit{3j}

symbol can be expressed as the finite sum …

4: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

►The

\mathit{6j}

symbol is defined by the following double sum of products of

\mathit{3j}

symbols: …where the summation is taken over all admissible values of the

m

’s and

m^{\prime}

’s for each of the four

\mathit{3j}

symbols; compare (34.2.2) and (34.2.3). … ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►where

{{}_{4}F_{3}}

is defined as in §16.2. …

5: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►

Row and Column Vectors

… ►and the corresponding transposed row vector of length

n

is … ►Two vectors

\mathbf{u}

and

\mathbf{v}

are orthogonal if … ►

Vector Norms

…

6: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

►For information on

12j,15j

,…, symbols, see Varshalovich et al. (1988, §10.12) and Yutsis et al. (1962, pp. 62–65 and 122–153).

7: 35.4 Partitions and Zonal Polynomials

… ►A partition

\kappa=(k_{1},\dots,k_{m})

is a vector of nonnegative integers, listed in nonincreasing order. Also,

|\kappa|

denotes

k_{1}+\dots+k_{m}

, the weight of

\kappa

;

\ell(\kappa)

denotes the number of nonzero

k_{j}

;

a+\kappa

denotes the vector

(a+k_{1},\dots,a+k_{m})

. … ►

35.4.1

{\left[a\right]_{\kappa}}=\frac{\Gamma_{m}\left(a+\kappa\right)}{\Gamma_{m}% \left(a\right)}=\prod_{j=1}^{m}{\left(a-\tfrac{1}{2}(j-1)\right)_{k_{j}}},

ⓘ

Defines:: ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(i), §35.4 and Ch.35

… ►

35.4.4

Z_{\kappa}\left(\boldsymbol{{0}}\right)=\begin{cases}1,&\kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}

ⓘ

Symbols:: $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.5

Z_{\kappa}\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}\right)=Z_{\kappa}\left(% \mathbf{T}\right),

\mathbf{H}\in\mathbf{O}(m)

ⓘ

Symbols:: $\in$ : element of, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…

8: 3.7 Ordinary Differential Equations

… ►and

\mathbf{b}(\tau,z)

is the vector ►

3.7.7

\mathbf{b}(\tau,z)=\begin{bmatrix}b_{1}(\tau,z)\\ b_{2}(\tau,z)\end{bmatrix},

ⓘ

Defines:: $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally) and $b_{\NVar{j}}(\NVar{\tau},\NVar{z})$ : vector (locally)
Permalink:: http://dlmf.nist.gov/3.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §3.7(ii), §3.7 and Ch.3

… ►Also let

\mathbf{w}

denote the

(2P+2)\times 1

vector …and

\mathbf{b}

the

(2P)\times 1

vector ►

3.7.12

\mathbf{b}=\left[b_{1}(\tau_{0},z_{0}),b_{2}(\tau_{0},z_{0}),b_{1}(\tau_{1},z_% {1}),b_{2}(\tau_{1},z_{1}),\ldots,b_{1}(\tau_{P-1},z_{P-1}),b_{2}(\tau_{P-1},z% _{P-1})\right]^{\rm T}.

ⓘ

Defines:: $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally) and $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally)
Symbols:: $\tau_{j}$ : change of variable and $P$ : partitioning point
Permalink:: http://dlmf.nist.gov/3.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §3.7(iii), §3.7 and Ch.3

…

9: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime

p

, the Legendre symbol

(n|p)

is defined as follows. If

p

divides

n

, then the value of

(n|p)

0

. … ►If an odd integer

P

has prime factorization

P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}

, then the Jacobi symbol

(n|P)

is defined by

(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}

, with

(n|1)=1

. The Jacobi symbol

(n|P)

is a Dirichlet character (mod

P

). …

10: 34.14 Tables

§34.14 Tables

… ►Some selected

\mathit{9j}

symbols are also given. Other tabulations for

\mathit{3j}

symbols are listed on pp. 11-12; for

\mathit{6j}

symbols on pp. 16-17; for

\mathit{9j}

symbols on p. …

Levi-Civita symbol for vectors

1—10 of 232 matching pages

1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

Levi-Civita Symbol

2: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

3: 34.2 Definition: 3 ⁢ j Symbol

§34.2 Definition: 3 ⁢ j Symbol

4: 34.4 Definition: 6 ⁢ j Symbol

§34.4 Definition: 6 ⁢ j Symbol

5: 1.2 Elementary Algebra

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

Row and Column Vectors

Vector Norms

6: 34.11 Higher-Order 3 ⁢ n ⁢ j Symbols

§34.11 Higher-Order 3 ⁢ n ⁢ j Symbols

7: 35.4 Partitions and Zonal Polynomials

8: 3.7 Ordinary Differential Equations

9: 27.9 Quadratic Characters

§27.9 Quadratic Characters

10: 34.14 Tables

§34.14 Tables

2: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

3: 34.2 Definition: $\mathit{3j}$ Symbol

§34.2 Definition: $\mathit{3j}$ Symbol

4: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

6: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols