Page 16 - Intro to Tensor Calculus

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We next sum on the index j which repeats itself in each term of equation (1.1.3). This gives

C i = e i11 A 1 B 1 + e i21 A 2 B 1 + e i31 A 3 B 1
+ e i12 A 1 B 2 + e i22 A 2 B 2 + e i32 A 3 B 2 (1.1.4)
+ e i13 A 1 B 3 + e i23 A 2 B 3 + e i33 A 3 B 3 .

Now we are left with i being a free index which can have any of the values of 1, 2or3. Letting i =1, then
letting i =2, and finally letting i = 3 produces the cross product components

C 1 = A 2 B 3 − A 3 B 2

C 2 = A 3 B 1 − A 1 B 3
C 3 = A 1 B 2 − A 2 B 1 .
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The cross product can also be expressed in the form A × B = e ijk A j B k b e i . This result can be verified by
summing over the indices i,j and k.

EXAMPLE 1.1-13. Show

for i, j, k =1, 2, 3
e ijk = −e ikj = e jki
Solution: The array ik j represents an odd number of transpositions of the indices ij k and to each
transposition there is a sign change of the e-permutation symbol. Similarly, jk i is an even transposition
of ij k and so there is no sign change of the e-permutation symbol. The above holds regardless of the
numerical values assigned to the indices i, j, k.

The e-δ Identity
An identity relating the e-permutation symbol and the Kronecker delta, which is useful in the simpli-
fication of tensor expressions, is the e-δ identity. This identity can be expressed in different forms. The
subscript form for this identity is

e ijk e imn = δ jm δ kn − δ jn δ km , i,j,k,m,n =1, 2, 3

where i is the summation index and j, k, m, n are free indices. A device used to remember the positions of
the subscripts is given in the figure 1.1-3.
The subscripts on the four Kronecker delta’s on the right-hand side of the e-δ identity then are read

(first)(second)-(outer)(inner).

This refers to the positions following the summation index. Thus, j, m are the first indices after the sum-
mation index and k, n are the second indices after the summation index. The indices j, n are outer indices
when compared to the inner indices k, m as the indices are viewed as written on the left-hand side of the
identity.