Maths - Tensor - Permutation Tensor

 

e123 = 0 if any of i, j, k are equal
= 1 if i,j, k are unequal and in cyclic order
= -1 if i,j, k are unequal and not in cyclic order

permutation matrix 3D

This can be defined by a determinant

e123=
ai1 ai2 ai3
aj1 aj2 aj3
ak1 ak2 ak3

This represents the cross product:

c = a × b

or in tensor notation:

ci = eijk a j b k

then expanding out gives:

c1 = e123 a 2 b 3 + e132 a 3 b 2 = a 2 b 3 - a 3 b 2
c2 = e231 a 3 b 1 + e213 a 1 b 3 = a 3 b 1 - a 1 b 3
c3 = e312 a 1 b 2 + e321 a 2 b 1 = a 1 b 2 - a 2 b 1

 

 


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