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 |
This can be defined by a determinant
e123= |
|
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