Symbol of Levi Civita

The Levi Civita symbol is a mathematical symbol that is used in tensor analysis . Named after the Italian mathematician Tullio Levi-Civita . Designated $\varepsilon _{ijk}$ ${\ displaystyle \ varepsilon _ {ijk}}$ $\ varepsilon_ {ijk}$ . Here is a symbol for three-dimensional space; for other dimensions, the number of indices changes (see below).

Other names:

Absolutely Antisymmetric Unit Tensor
Fully Antisymmetric Unit Tensor
Absolutely skew-symmetric object
The Levi-Civita tensor (the Levi-Civita symbol is a component notation of this tensor).
The skew-symmetric Kronecker symbol (this term was used in the textbook on tensor calculus of Akivis and Goldberg)

Content

Definition

Image of a Levi Civita symbol.

In three-dimensional space, in the right orthonormal basis (or even in the right basis with the unit determinant of the metric), the Levi-Civita symbol is defined as follows:

\varepsilon _{ijk}={\begin{cases}+1&P(i,j,k)=+1\\-1&P(i,j,k)=-1\\0&i=j\bigvee j=k\bigvee k=i\end{cases}}

{\ displaystyle \ varepsilon _ {ijk} = {\ begin {cases} + 1 & P (i, j, k) = + 1 \\ - 1 & P (i, j, k) = - 1 \\ 0 & i = j \ bigvee j = k \ bigvee k = i \ end {cases}}}

that is, for an even permutation P (i, j, k) is equal to 1 (for triples (1,2,3), (2,3,1), (3,1,2)), for an odd permutation P (i, j, k) is −1 (for triples (3,2,1), (1,3,2), (2,1,3)), and in other cases it is equal to zero, when repeated. For components $\ \varepsilon _{ijk}$ ${\ displaystyle \ \ varepsilon _ {ijk}}$ opposite numbers are taken in the left basis.

For the general case (arbitrary oblique coordinates with the right orientation of the basis vectors), this definition usually changes to

\varepsilon _{ijk}={\begin{cases}+{\sqrt {g}}&P(i,j,k)=+1\\-{\sqrt {g}}&P(i,j,k)=-1\\0&i=j\bigvee j=k\bigvee k=i\end{cases}}

{\ displaystyle \ varepsilon _ {ijk} = {\ begin {cases} + {\ sqrt {g}} & P (i, j, k) = + 1 \\ - {\ sqrt {g}} & P (i, j , k) = - 1 \\ 0 & i = j \ bigvee j = k \ bigvee k = i \ end {cases}}}

Where $\ g$ ${\ displaystyle \ g}$ - determinant of the metric tensor matrix $\ g_{ij}$ ${\ displaystyle \ g_ {ij}}$ representing the square of the volume of the parallelepiped stretched over the basis. For components $\ \varepsilon _{ijk}$ ${\ displaystyle \ \ varepsilon _ {ijk}}$ opposite numbers are taken in the left basis.

This set of components $\varepsilon _{ijk}$ ${\ displaystyle \ varepsilon _ {ijk}}$ is a (true) tensor . If, as is sometimes done in the literature, as a definition $\varepsilon _{ijk}$ ${\ displaystyle \ varepsilon _ {ijk}}$ use the above formulas for any - both right and left - coordinate systems, the resulting set of numbers will represent a pseudo tensor .

Wherein $\varepsilon ^{ijk}$ ${\ displaystyle \ varepsilon ^ {ijk}}$ will be the same but with a replacement $\ {\sqrt {g}}$ ${\ displaystyle \ {\ sqrt {g}}}$ on $\ 1/{\sqrt {g}}$ ${\ displaystyle \ 1 / {\ sqrt {g}}}$ .

$\varepsilon _{ijk}$ ${\ displaystyle \ varepsilon _ {ijk}}$ can also be defined as a mixed product of basis vectors in which the symbol is used:

\varepsilon _{ijk}=\left[{\vec {e}}_{i}{\vec {e}}_{j}{\vec {e}}_{k}\right]

{\ displaystyle \ varepsilon _ {ijk} = \ left [{\ vec {e}} _ {i} {\ vec {e}} _ {j} {\ vec {e}} _ {k} \ right]}

.

This definition is for any, right or left basis, since the difference in sign for the left and right bases is in the mixed work. The absolute value of each nonzero component is equal to the volume of the parallelepiped stretched to the basis $\ \{{\vec {e_{i}}}\}$ ${\ displaystyle \ \ {{\ vec {e_ {i}}} \}}$ . The tensor, as expected, is antisymmetric for any pair of indices. The definition is equivalent to the above.

Sometimes they use an alternative definition of the Levi-Civita symbol without a multiplier ${\sqrt {g}}\$ ${\ displaystyle {\ sqrt {g}} \}$ in any bases (i.e., such that all its components are always equal to ± 1 or 0, as in our definition for orthonormal bases ). In this case, it is not in itself a representation of the tensor. Multiplied by ${\sqrt {g}}\$ ${\ displaystyle {\ sqrt {g}} \}$ object (matching with $\varepsilon _{ijk}$ ${\ displaystyle \ varepsilon _ {ijk}}$ in our definition, which is a tensor) in this case is denoted by another letter and is usually called an element of volume . Here we follow the definition of Levi-Civita. (This remark is valid not only for three-dimensional space, but also for any dimension).

Geometric meaning

As can be seen already from the definition through a mixed product, the Levi-Civita symbol is associated with an oriented volume and an oriented area, represented as a vector.

In three-dimensional (Euclidean) space, the mixed product of three vectors

V=\varepsilon _{ijk}a^{i}b^{j}c^{k}

{\ displaystyle V = \ varepsilon _ {ijk} a ^ {i} b ^ {j} c ^ {k}}

Is an oriented volume (a pseudoscalar whose modulus is equal to the volume, and the sign depends on the orientation of the triple of vectors) of a parallelepiped spanned by three vectors ${\vec {a}}$ ${\ displaystyle {\ vec {a}}}$ , ${\vec {b}}$ ${\ displaystyle {\ vec {b}}}$ and ${\vec {c}}$ ${\ displaystyle {\ vec {c}}}$

Vector product of two vectors

S_{i}=\varepsilon _{ijk}a^{j}b^{k}

{\ displaystyle S_ {i} = \ varepsilon _ {ijk} a ^ {j} b ^ {k}}

Is the oriented area of the parallelogram whose sides are vectors ${\vec {a}}$ ${\ displaystyle {\ vec {a}}}$ and ${\vec {b}}$ ${\ displaystyle {\ vec {b}}}$ represented by a pseudovector whose length is equal to the area and the direction is orthogonal to the plane of the parallelogram.

This meaning is preserved for any dimension of the space n , if, of course, we take $\varepsilon$ ${\ displaystyle \ varepsilon}$ with the corresponding number of indices, by volume we mean n- dimensional volume, and by area - ( n −1) -dimensional (hyper-) area. Moreover, naturally, the corresponding formula includes n and ( n −1) vectors - factors. For example, for 4-dimensional (Euclidean) space:

V=\varepsilon _{ijkm}a^{i}b^{j}c^{k}d^{m},

{\ displaystyle V = \ varepsilon _ {ijkm} a ^ {i} b ^ {j} c ^ {k} d ^ {m},}

S_{i}=\varepsilon _{ijkm}a^{j}b^{k}c^{m}.

{\ displaystyle S_ {i} = \ varepsilon _ {ijkm} a ^ {j} b ^ {k} c ^ {m}.}

Properties

The determinant of a matrix A of size 3 × 3 can be written (here we mean a standard , and therefore an orthonormal basis):
${\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}=\sum _{i,j,k=1}^{3}\varepsilon _{ijk}a_{i}b_{j}c_{k}.$ ${\ displaystyle {\ begin {vmatrix} a_ {1} & a_ {2} & a_ {3} \\ b_ {1} & b_ {2} & b_ {3} \\ c_ {1} & c_ {2} & c_ {3} \ \\ end {vmatrix}} = \ sum _ {i, j, k = 1} ^ {3} \ varepsilon _ {ijk} a_ {i} b_ {j} c_ {k}.}$
The vector product of two spatial vectors is written through this symbol:
${\vec {a}}\times {\vec {b}}=\sum _{i,j,k=1}^{3}\varepsilon _{ijk}{\vec {e}}^{\ i}a^{j}b^{k}={\vec {c}}$ ${\ displaystyle {\ vec {a}} \ times {\ vec {b}} = \ sum _ {i, j, k = 1} ^ {3} \ varepsilon _ {ijk} {\ vec {e}} ^ {\ i} a ^ {j} b ^ {k} = {\ vec {c}}}$ where $c_{i}=\sum _{j,k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}$ ${\ displaystyle c_ {i} = \ sum _ {j, k = 1} ^ {3} \ varepsilon _ {ijk} a ^ {j} b ^ {k}}$ - its components, and $\ {\vec {e}}^{\ i}$ ${\ displaystyle \ {\ vec {e}} ^ {\ i}}$ are basis vectors.
Mixed product of vectors too:
$\left[{\vec {a}}{\vec {b}}{\vec {c}}\right]=\sum _{i,j,k=1}^{3}\varepsilon _{ijk}a^{i}b^{j}c^{k}.$ ${\ displaystyle \ left [{\ vec {a}} {\ vec {b}} {\ vec {c}} \ right] = \ sum _ {i, j, k = 1} ^ {3} \ varepsilon _ {ijk} a ^ {i} b ^ {j} c ^ {k}.}$
In the following formula $\delta$ ${\ displaystyle \ delta}$ denotes the Kronecker symbol :
$\varepsilon _{ijk}\varepsilon ^{lmn}={\begin{vmatrix}\delta _{i}^{l}&\delta _{i}^{m}&\delta _{i}^{n}\\\delta _{j}^{l}&\delta _{j}^{m}&\delta _{j}^{n}\\\delta _{k}^{l}&\delta _{k}^{m}&\delta _{k}^{n}\\\end{vmatrix}}.$ ${\ displaystyle \ varepsilon _ {ijk} \ varepsilon ^ {lmn} = {\ begin {vmatrix} \ delta _ {i} ^ {l} & \ delta _ {i} ^ {m} & \ delta _ {i} ^ {n} \\\ delta _ {j} ^ {l} & \ delta _ {j} ^ {m} & \ delta _ {j} ^ {n} \\\ delta _ {k} ^ {l} & \ delta _ {k} ^ {m} & \ delta _ {k} ^ {n} \\\ end {vmatrix}}.}$
In the case of summation over the general index

\sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}

{\ displaystyle \ sum _ {i = 1} ^ {3} \ varepsilon _ {ijk} \ varepsilon _ {imn} = \ delta _ {jm} \ delta _ {kn} - \ delta _ {jn} \ delta _ {km}}

In the case of two general indices $i,\;j$ ${\ displaystyle i, \; j}$ , the tensor collapses as follows:

\sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}

{\ displaystyle \ sum _ {i = 1} ^ {3} \ sum _ {j = 1} ^ {3} \ varepsilon _ {ijk} \ varepsilon _ {ijn} = 2 \ delta _ {kn}}

(Everywhere here in the case of an orthonormal basis, all indices can simply be rewritten as lower ones.)

Generalization to the case of n measurements

The Levi-Civita symbol can be easily generalized to any number of dimensions greater than one, if we use the definition through the parity of permutations of indices:

$\varepsilon _{ijk\ell \dots }=\left\{{\begin{matrix}~\\~\\~\end{matrix}}\right.$ ${\ displaystyle \ varepsilon _ {ijk \ ell \ dots} = \ left \ {{\ begin {matrix} ~ \\ ~ \\ ~ \ end {matrix}} \ right.}$	$+{\sqrt {g}},$ ${\ displaystyle + {\ sqrt {g}},}$ if a $(i,j,k,\ell ,\dots )$ ${\ displaystyle (i, j, k, \ ell, \ dots)}$ there is an even permutation of the set $(1,2,3,4,\dots )\;;$ ${\ displaystyle (1,2,3,4, \ dots) \ ;;}$
	$-{\sqrt {g}},$ ${\ displaystyle - {\ sqrt {g}},}$ if a $(i,j,k,\ell ,\dots )$ ${\ displaystyle (i, j, k, \ ell, \ dots)}$ there is an odd permutation of the set $(1,2,3,4,\dots )\;;$ ${\ displaystyle (1,2,3,4, \ dots) \ ;;}$
	$0$ ${\ displaystyle 0}$ if at least two indices match.

That is, it is equal to the sign (signum) of the permutation , multiplied by the root of the determinant of the metric ${\sqrt {g}}={\sqrt {\det {\{g_{ij}\}}}}$ ${\ displaystyle {\ sqrt {g}} = {\ sqrt {\ det {\ {g_ {ij} \}}}}}$ in the case when the indices take values that implement the permutation of the set $(1,2,3,\dots ,n)$ ${\ displaystyle (1,2,3, \ dots, n)}$ , and in other cases zero. (As you can see, the number of indices is equal to the dimension of space $n$ ${\ displaystyle n}$ .)

In pseudo-Euclidean spaces , if the signature of the metric is such that $g<0$ ${\ displaystyle g <0}$ , as a rule they take instead $-g$ ${\ displaystyle -g}$ to ${\sqrt {g}}$ ${\ displaystyle {\ sqrt {g}}}$ It turned out real.

In all dimensions where the Levi-Civita symbol is defined, it represents a tensor (meaning mainly what needs to be ensured so that the number of symbol indices matches the dimension of the space). In addition, as can be seen from the above, some difficulties with the usual definition of the Levi-Civita symbol can be in spaces where the metric tensor is not defined, or, say, $\det {\{g_{ij}\}}=0$ ${\ displaystyle \ det {\ {g_ {ij} \}} = 0}$ or $\det {\{g^{ij}\}}=0$ ${\ displaystyle \ det {\ {g ^ {ij} \}} = 0}$ .

It can be shown that for $n$ ${\ displaystyle n}$ measurements are performed properties similar to three-dimensional:

$\sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon ^{ijk\dots }=n!$ ${\ displaystyle \ sum _ {i, j, k, \ dots = 1} ^ {n} \ varepsilon _ {ijk \ dots} \ varepsilon ^ {ijk \ dots} = n!}$

- due to the fact that there is $n!$ ${\ displaystyle n!}$ set permutations $(1,2,3,\dots ,n)$ ${\ displaystyle (1,2,3, \ dots, n)}$ , and therefore, as many nonzero components $\varepsilon$ ${\ displaystyle \ varepsilon}$ with $n$ ${\ displaystyle n}$ indexes.

$\varepsilon _{ijk\dots }\varepsilon ^{pqr\dots }={\begin{vmatrix}\delta _{i}^{p}&\delta _{i}^{q}&\delta _{i}^{r}&\dots \\\delta _{j}^{p}&\delta _{j}^{q}&\delta _{j}^{r}&\dots \\\delta _{k}^{p}&\delta _{k}^{q}&\delta _{k}^{r}&\dots \\\vdots &\vdots &\vdots &\ddots \\\end{vmatrix}}.$ ${\ displaystyle \ varepsilon _ {ijk \ dots} \ varepsilon ^ {pqr \ dots} = {\ begin {vmatrix} \ delta _ {i} ^ {p} & \ delta _ {i} ^ {q} & \ delta _ {i} ^ {r} & \ dots \\\ delta _ {j} ^ {p} & \ delta _ {j} ^ {q} & \ delta _ {j} ^ {r} & \ dots \\ \ delta _ {k} ^ {p} & \ delta _ {k} ^ {q} & \ delta _ {k} ^ {r} & \ dots \\\ vdots & \ vdots & \ vdots & \ ddots \\ \ end {vmatrix}}.}$

After expanding the determinant, a multiplier appears $n!$ ${\ displaystyle n!}$ and simplifications are made in the corresponding Kronecker symbols.

The pseudoscalar product of two vectors in two-dimensional space:
${{\vec {a}}\vee {\vec {b}}}=\sum _{i,j=1}^{2}\varepsilon _{ij}a^{i}b^{j}$ ${\ displaystyle {{\ vec {a}} \ vee {\ vec {b}}} = \ sum _ {i, j = 1} ^ {2} \ varepsilon _ {ij} a ^ {i} b ^ { j}}$

Matrix determinant $A$ ${\ displaystyle A}$ the size $n\times n$ ${\ displaystyle n \ times n}$ can be conveniently recorded using $n$ ${\ displaystyle n}$ -dimensional symbol of Levi-Civita
$\det {A}=\sum _{i,j,k,\ldots =1}^{n}\varepsilon _{ijk\ldots }A_{1i}A_{2j}A_{3k}\cdots =\sum _{i_{1},i_{2},i_{3},\ldots ,i_{n}=1}^{n}\varepsilon _{i_{1}i_{2}i_{3}\cdots i_{n}}A_{1i_{1}}A_{2i_{2}}A_{3i_{3}}\cdots A_{ni_{n}}$ ${\ displaystyle \ det {A} = \ sum _ {i, j, k, \ ldots = 1} ^ {n} \ varepsilon _ {ijk \ ldots} A_ {1i} A_ {2j} A_ {3k} \ cdots = \ sum _ {i_ {1}, i_ {2}, i_ {3}, \ ldots, i_ {n} = 1} ^ {n} \ varepsilon _ {i_ {1} i_ {2} i_ {3} \ cdots i_ {n}} A_ {1i_ {1}} A_ {2i_ {2}} A_ {3i_ {3}} \ cdots A_ {ni_ {n}}}$

which is, in fact, simply rewritten with the help of this symbol the definition of a determinant (one of the most common). Here, the basis is assumed to be standard, and nonzero components $\varepsilon _{ijk\ldots }$ ${\ displaystyle \ varepsilon _ {ijk \ ldots}}$ take values here $\pm 1$ ${\ displaystyle \ pm 1}$ .

direct $n$ ${\ displaystyle n}$ -dimensional generalization of a vector product $n-1$ ${\ displaystyle n-1}$ pieces (' $n$ ${\ displaystyle n}$ -dimensional) vectors:

{\vec {p}}={{\vec {a}}\times {\vec {b}}\times {\vec {c}}\cdots }=\sum _{i,j,k,m,\ldots =1}^{n}\varepsilon _{ijkm\ldots }{\vec {f}}^{i}a^{j}b^{k}c^{m}\cdots

{\ displaystyle {\ vec {p}} = {{\ vec {a}} \ times {\ vec {b}} \ times {\ vec {c}} \ cdots} = \ sum _ {i, j, k , m, \ ldots = 1} ^ {n} \ varepsilon _ {ijkm \ ldots} {\ vec {f}} ^ {i} a ^ {j} b ^ {k} c ^ {m} \ cdots}

,

Where $p_{i}=\sum _{j,k,m,\ldots =1}^{n}\varepsilon _{ijkm\ldots }a^{j}b^{k}c^{m}\cdots$ ${\ displaystyle p_ {i} = \ sum _ {j, k, m, \ ldots = 1} ^ {n} \ varepsilon _ {ijkm \ ldots} a ^ {j} b ^ {k} c ^ {m} \ cdots}$ - its components, and ${\vec {f}}^{\ i}$ ${\ displaystyle {\ vec {f}} ^ {\ i}}$ - basis vectors. (Here, for brevity, the expression for covariant components and the expansion in the dual basis are written).

direct $n$ ${\ displaystyle n}$ -dimensional generalization of a mixed product $n$ ${\ displaystyle n}$ pieces ( $n$ ${\ displaystyle n}$ -dimensional) vectors:
$\left[{\vec {a}}{\vec {b}}{\vec {c}}\cdots \right]=\sum _{i,j,k,\ldots =1}^{n}\varepsilon _{ijk\ldots }a^{i}b^{j}c^{k}\cdots$ ${\ displaystyle \ left [{\ vec {a}} {\ vec {b}} {\ vec {c}} \ cdots \ right] = \ sum _ {i, j, k, \ ldots = 1} ^ { n} \ varepsilon _ {ijk \ ldots} a ^ {i} b ^ {j} c ^ {k} \ cdots}$

Indexless record (for n measurements)

In a non-index tensor notation, the Levi-Civita symbol is replaced by a duality operator called the Hodge asterisk , or simply the asterisk operator:

(*\eta )_{i_{1},i_{2},\ldots ,i_{n-k}}={\frac {1}{k!}}\eta ^{j_{1},\ldots ,j_{k}}\varepsilon _{j_{1},\ldots ,j_{k},i_{1},\ldots ,i_{n-k}}

{\ displaystyle (* \ eta) _ {i_ {1}, i_ {2}, \ ldots, i_ {nk}} = {\ frac {1} {k!}} \ eta ^ {j_ {1}, \ ldots, j_ {k}} \ varepsilon _ {j_ {1}, \ ldots, j_ {k}, i_ {1}, \ ldots, i_ {nk}}}

(for an arbitrary tensor $\eta ,$ ${\ displaystyle \ eta,}$ given the Einstein summation rule ).

Links

Hermann R. (ed.), Ricci and Levi-Civita's tensor analysis papers , (1975) Math Sci Press, Brookline (symbol definition - see page 31).
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation , (1970) WH Freeman, New York; ISBN 0-7167-0344-0 . (See paragraph 3.5 for a review of the application of tensors in general relativity .)
Russian translation: C. Mizner, C. Thorne, J. Wheeler, Gravity , (1977) Moscow, “Mir” (See Levi-Civita tensor according to the index).
Dimitrienko Yu.I., Tensor calculus , M.: Higher school, 2001, 575 p.