Levi-Civita Guio.

Levi-Civita symbol and cross product vector/tensor Patrick Guio $Id: levi-civita.tex,v 1.9 2001/08/29 15:53:02 patricg Exp $ Definitions The Levi-Civita symbol is a tensor of rank three and is defined by (1) The Levi-Civita symbol is anti-symmetric on each pair of indexes. The determinant of a matrix with elements can be written in term of as (2) Note the compact notation where the summation over the spatial directions is dropped. It is this one that is in use. 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) or for each coordinate (5)Levi-Civita symbol and cross product vector/tensor http://folk.uio.no/patricg/teaching/a112/levi-civita/ 1 of 5 9/1/2011 7:59 AM Properties The Levi-Civita tensor has components. components are equal to . components are equal to . components are equal to . Identities The product of two Levi-Civita symbols can be expressed as a function of the Kronecker's symbol (6) Setting gives (7) proof Setting and gives (8) Setting , and gives (9) ThereforeLevi-Civita symbol and cross product vector/tensor http://folk.uio.no/patricg/teaching/a112/levi-civita/ 2 of 5 9/1/2011 7:59 AM (10) proof In the same way Properties The cross product is a special vector. If we transform both vectors by a reflection transformation, for example a central symmetry by the origin, i.e. , the cross product vector is conserved. proofLevi-Civita symbol and cross product vector/tensor http://folk.uio.no/patricg/teaching/a112/levi-civita/ 3 of 5 9/1/2011 7:59 AM 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. proof The correct or consistent approach of calculating the cross product vector from the tensor is the so called index contraction (11) proofLevi-Civita symbol and cross product vector/tensor http://folk.uio.no/patricg/teaching/a112/levi-civita/ 4 of 5 9/1/2011 7:59 AM In 4 dimensions, the cross product tensor is thus written (12) This tensor has 6 independent components. There should be 4 components for a 4 dimensions vector, therefore it cannot be represented as a vector. 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 2001-08-29Levi-Civita symbol and cross product vector/tensor http://folk.uio.no/patricg/teaching/a112/levi-civita/ 5 of 5 9/1/2011 7:59 AM