According to the Einstein notation, the summation symbol may be omitted.
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 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.
The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations:
("contracted epsilon identity")
(In Einstein notation, the duplication of the i index implies the sum on i. The previous is then noted: )
Γενίκευση σε n διαστάσεις[]
The Levi-Civita symbol can be generalized to higher dimensions:
Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.
Furthermore, for any n the property
follows from the facts that (a) every permutation is either even or odd, (b) (+1)2 = (-1)2 = 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 dimensions one can write the product of two Levi-Civita symbols as:
.
Now we can contract indices. This will add a factor of to the determinant and we need to omit the relevant Kronecker delta.
Ιδιότητες[]
(in these examples, superscripts should be considered equivalent with subscripts)
1. When , we have for all in ,
, (1)
, (2)
. (3)
2. When , we have for all in
, (4)
, (5)
. (6)
Αποδείξεις[]
For equation 1, both sides are antisymmetric with respect of and . We therefore only need to consider the case and . By substitution, we see that the equation holds for , i.e., for and . (Both sides are then one). Since the equation is antisymmetric in and , any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of and . Using equation 1, we have for equation 2
.
Here we used the Einstein summation convention with going from to . Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when . Indeed, if , then one can not choose and such that both permutation symbols on the left are nonzero. Then, with fixed, there are only two ways to choose and from the remaining two indices. For any such indices, we have (no summation), and the result follows. Property (5) follows since and for any distinct indices in , we have (no summation).
2. If and are vectors in (represented in some right hand oriented orthonormal basis), then the th component of their cross product equals
For instance, the first component of is . From the above expression for the cross product, it is clear that . Further, if is a vector like and , then the triple scalar product equals
From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any adjacent arguments. For example, .
3. Suppose is a vector field defined on some open set of with Cartesian coordinates . Then the th component of the curl of equals
Notation[]
A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, for an n x n matrix, M,
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (See section 3.5 for a review of tensors in general relativity).
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