Page 105 - Intro to Tensor Calculus

100

Dual Tensors
a
The e-permutation symbol is often used to generate new tensors from given tensors. For T i 1 i 2 ...i m
skew-symmetric tensor, we define the tensor

1
T ˆ j 1j 2 ...j n−m = e j 1 j 2 ...j n−m i 1 i 2 ...i m T i 1 i 2 ...i m m ≤ n (1.3.81)
m!
. Note that the e-permutation symbol or alternating tensor has
as the dual tensor associated with T i 1 i 2 ...i m
a weight of +1 and consequently the dual tensor will have a higher weight than the original tensor.
The e-permutation symbol has the following properties
e i 1 i 2 ...i N e i 1 i 2 ...i N = N!

e i 1 i 2 ...i N e j 1 j 2 ...j N = δ i 1 i 2 ...i N
j 1 j 2 ...j N
(1.3.82)
e j 1 j 2 ...j m i 1 i 2 ...i N−m =(N − m)!δ j 1 j 2 ...j m
e k 1 k 2 ...k m i 1 i 2 ...i N−m
k 1 k 2 ...k m
δ j 1 j 2 ...j m T j 1 j 2 ...j m = m!T k 1 k 2 ...k m .
k 1 k 2 ...k m
Using the above properties we can solve for the skew-symmetric tensor in terms of the dual tensor. We find
1
= T ˆ j 1 j 2 ...j n−m . (1.3.83)
T i 1 i 2 ...i m e i 1 i 2 ...i m j 1 j 2 ...j n−m
(n − m)!
For example, if A ij i, j =1, 2, 3 is a skew-symmetric tensor, we may associate with it the dual tensor

1 ijk
i
V = e A jk ,
2!
which is a first order tensor or vector. Note that A ij has the components
 
0 A 12 A 13
 −A 12 0 A 23  (1.3.84)
−A 13 −A 23 0
~
and consequently, the components of the vector V are
1 2 3
(V ,V ,V )= (A 23 ,A 31 ,A 12 ). (1.3.85)

Note that the vector components have a cyclic order to the indices which comes from the cyclic properties
of the e-permutation symbol.
As another example, consider the fourth order skew-symmetric tensor A ijkl , i,j,k,l =1,... ,n.We can
associate with this tensor any of the dual tensor quantities
1 ijkl
V = e A ijkl
4!
1 ijklm
i
V = e A jklm
4!
1 ijklmn
ij
V = e A klmn (1.3.86)
4!
1 ijklmnp
ijk
V = e A lmnp
4!
1 ijklmnpr
ijkl
V = e A mnpr
4!
Applications of dual tensors can be found in section 2.2.