Page 12 - Intro to Tensor Calculus
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The definition of permutations can be used to define the e-permutation symbol.
Definition: (e-Permutation symbol or alternating tensor)
The e-permutation symbol is defined
1 if ijk ... l is an even permutation of the integers 123 ...n
e ijk...l = e ijk...l = −1 if ijk ... l is an odd permutation of the integers 123 ...n
0 in all other cases
EXAMPLE 1.1-5. Find e 612453 .
Solution: To determine whether 612453 is an even or odd permutation of 123456 we write down the given
numbers and below them we write the integers 1 through 6. Like numbers are then connected by a line and
we obtain figure 1.1-2.
Figure 1.1-2. Permutations of 123456.
In figure 1.1-2, there are seven intersections of the lines connecting like numbers. The number of
intersections is an odd number and shows that an odd number of transpositions must be performed. These
results imply e 612453 = −1.
Another definition used quite frequently in the representation of mathematical and engineering quantities
is the Kronecker delta which we now define in terms of both subscripts and superscripts.
Definition: (Kronecker delta) The Kronecker delta is defined:
1 if i equals j
j
δ ij = δ =
i
0 if i is different from j
