Tensors for Physics

54 4 Epsilon-Tensor

This 2Dε-tensor can also be expressed in matrix notation:

εij=

(

01

− 10

)

. (4.28)

In 3D, a corresponding notation requires a three dimensional ‘3× 3 ×3-matrix’.

4.1 Exercise: 2D Dual Relation in Complex Number Notation

Let the two 2D vectors(x 1 ,y 1 )and(x 2 ,y 2 )be expressed in terms of the complex
numbersz 1 =x 1 +iy 1 andz 2 =x 2 +iy 2. Write the dual relation corresponding to
(4.25) in terms of the complex numbersz 1 andz 2. How about the scalar product of
these 2D vectors?
Hint: the complex conjugate ofz=x+iyisz∗=x−iy.