|
:<math>
\varepsilon_{ijkl} = \frac{\left( j-i \right)\left( k-i \right)\left( l-i \right)\left( k-j \right)\left( l-j \right)\left( l-k \right)}{12} = \frac{\left( i-j \right)\left( i-k \right)\left( i-l \right)\left( j-k \right)\left( j-l \right)\left( k-l \right)}{12}.
</math>
[[სურათი:LeviCivitaTensor.jpg|thumb|250px|მარცხენა ორიენტაციის კოორდინატთა სისტემაში ლევი-ჩივიტას სიმბოლოს ილუსტრაცია. ცარიელი კუბი აღნიშნავს 0-ს, წითელი +1-ს, ხოლო ლურჯი -1-ს.]]
მაგალითად, [[წრფივი ალგებრა|წრფივ ალგებრაში]] 3×3 A მატრიცის [[დეტერმინანტი]] მოიცემა შემდეგი ფორმულით:
:<math>
</math>
ხოლო ორი [[ვექტორი]]ს [[ვექტორული ნამრავლი]] ჩაიწერება შემდეგნაირად
(and similarly for a square matrix of general size, see below)
and the [[cross product]] of two [[Vector (geometry)|vector]]s can be written as a determinant:
:<math>
\mathbf{a \times b} =
= \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} \mathbf{e_i} a_j b_k
</math>
or more simply:
ან უფრო მარტივად:
:<math>
\mathbf{a \times b} = \mathbf{c},\ c_i = \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} a_j b_k.
</math>
===კავშირი კრონეკერის სიმბოლოსთან===
According to the [[Einstein notation]], the summation symbols may be omitted.
ლევი-ჩივიტას სიმბოლო დაკავშირებულია [[კრონეკერის სიმბოლო]]სთან. სამგანზომილებიან სივრცეში ეს კავშირი შემდეგი ფორმულებით მოიცემა:
The [[tensor]] whose components in an [[orthonormal basis]] are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the '''permutation tensor'''. It is actually a [[pseudotensor]] because under an orthogonal transformation of [[jacobian matrix and determinant|jacobian determinant]] −1 (i.e., a rotation composed with a reflection), it acquires a minus sign. Because the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a [[pseudovector]], not a vector.
Note that under a general coordinate change, the components of the permutation tensor get multiplied by the jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.
===Relation to Kronecker delta===
The Levi-Civita symbol is related to the [[Kronecker delta]]. In three dimensions, the relationship is given by the following equations:
:<math>
\delta_{jl} & \delta_{jm}& \delta_{jn}\\
\delta_{kl} & \delta_{km}& \delta_{kn}\\
\end{bmatrix},
</math>
:::<math> = \delta_{il}\left( \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\right) - \delta_{im}\left( \delta_{jl}\delta_{kn} - \delta_{jn}\delta_{kl} \right) + \delta_{in} \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \right) \,</math>
:<math>
\sum_{i=1}^3 \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}
</math> ("contracted epsilon identity")
(In [[Einstein notation]], the duplication of the i index implies the sum on i. The previous is then noted: <math> \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}</math>)
და
:<math>
\sum_{i=1}^3 \sum_{j=1}^3 \varepsilon_{ijk}\varepsilon_{ijn} = 2\delta_{kn}
</math>
===Generalization to ''n'' dimensions===
The Levi-Civita symbol can be generalized to higher dimensions:
:<math>\varepsilon_{ijkl\dots} =
\begin{cases}
+1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\
-1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\
0 & \mbox{otherwise}
\end{cases}
Thus, it is the [[Even and odd permutations|sign of the permutation]] in the case of a permutation, and zero otherwise.
The generalized formula is:
:<math>
\varepsilon_{a_1 a_2 a_3 \ldots a_n} = \prod_{i=1}^{n-1} \left( \frac{1}{i!} \ \prod_{j=i+1}^n ( a_j-a_i ) \right)
</math>
where n is the dimension (rank).
For any ''n'' the property
:<math>
\sum_{i,j,k,\dots=1}^n \varepsilon_{ijk\dots}\varepsilon_{ijk\dots} = n!
</math>
follows from the facts that (a) every permutation is either even or odd, (b) (+1)<sup>2</sup> = (-1)<sup>2</sup> = 1, and (c) the permutations of any ''n''-element set number exactly ''n''!.
In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the [[Hodge dual]].
In general ''n'' dimensions one can write the product of two Levi-Civita symbols as:
:<math> \varepsilon_{i_1 i_2 \dots i_n} \varepsilon_{j_1 j_2 \dots j_n} = \det \begin{bmatrix}
\delta_{i_1 j_1} & \delta_{i_1 j_2} & \dots & \delta_{i_1 j_n} \\
\delta_{i_2 j_1} & \delta_{i_2 j_2} & \dots & \delta_{i_2 j_n} \\
\vdots & \vdots & \ddots & \vdots \\
\delta_{i_n j_1} & \delta_{i_n j_2} & \dots & \delta_{i_n j_n} \\
\end{bmatrix} </math>.
==Properties==
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