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

 

 


metadata block
see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover Tensor Analysis.

Terminology and Notation

Specific to this page here:

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.