Chapter 4
Epsilon-Tensor
Abstract Thethirdrankepsilon-tensorisusedtoformulatethedualrelationbetween
an antisymmetric second rank tensor and a vector or vice versa, in three-dimensional
space. In this chapter, the properties of this isotropic tensor are presented. From the
rules for the multiplication of two of these tensors follow relations for the scalar
product of two vector products and the double vector product. Some applications
are presented, involving the orbital angular momentum, the torque, the motion on
a circle and on a screw curve, as well as the Lorentz force. The dual relation in
two-dimensional space is discussed.
4.1 Definition, Properties.
4.1.1 Link with Determinants
The dual relation between an antisymmetric second rank tensor and a vector, as well
as the properties of the vector product can be formulated more efficiently with the
help of the third rankepsilon-tensor, which is also calledLevi-Civita tensor.Itis
defined by
εμνλ:=∣ ∣ ∣ ∣ ∣ ∣
δ 1 μδ 1 νδ 1 λ
δ 2 μδ 2 νδ 2 λ
δ 3 μδ 3 νδ 3 λ∣ ∣ ∣ ∣ ∣ ∣
. (4.1)
This implies
1 ,μνλ= 123 , 231 , 312
εμνλ=− 1 ,μνλ= 213 , 132 , 321
0 ,μνλ=else, (4.2)or, equivalently,ε 123 =ε 231 =ε 312 =1,ε 213 =ε 132 =ε 321 =−1, andε...=0for
all other combinations of subscripts.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_4
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