**Patrick Guio**

`$Id: levi-civita.tex,v 1.9 2001/08/29 15:53:02 patricg Exp $`

The Levi-Civita symbol
is a tensor of rank three and
is defined by

(1) |

The determinant of a matrix with elements can be written
in term of
as

(2) |

Note that the Levi-Civita symbol can therefore be expressed as the
determinant, or mixed triple product, of any of the unit vectors
of a normalised and direct
orthogonal frame of reference.

(3) |

Now we can define by analogy to the definition of the determinant an
additional type of product, the vector product or simply cross product

(4) |

(5) |

- The Levi-Civita tensor has components.
- components are equal to .
- components are equal to .
- components are equal to .

The product of two Levi-Civita symbols can be expressed as a function
of the Kronecker's symbol

(6) |

Setting gives

(7) |

Setting and gives

(8) |

Setting , and gives

(9) |

Therefore

(10) |

In the same way

The cross product does not have the same properties as an ordinary vector. Ordinary vectors are called polar vectors while cross product vector are called axial (pseudo) vectors. In one way the cross product is an artificial vector.

Actually, there does not exist a cross product vector in space with more than 3 dimensions. The fact that the cross product of 3 dimensions vector gives an object which also has 3 dimensions is just pure coincidence.

The cross product in 3 dimensions is actually a tensor of rank 2 with 3 independent coordinates.

The correct or consistent approach of calculating the cross product vector
from the tensor
is
the so called index contraction

(11) |

In 4 dimensions, the cross product tensor is thus written

(12) |

More generally, if is the dimension of the vector, the cross product tensor is a tensor of rank 2 with independent components.

The cross product is connected to rotations and has a structure which also
looks like rotations, called a simplectic structure.

Patrick Guio 2001-08-29