Page 7 - Intro to Tensor Calculus

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the Kronecker delta symbol δ ij , defined by δ ij =1 if i = j and δ ij =0 for i 6= j,with i, j ranging over the
values 1,2,3, represents the 9 quantities

δ 11 =1 δ 12 =0 δ 13 =0

δ 21 =0 δ 22 =1 δ 23 =0
δ 31 =0 δ 32 =0 δ 33 =1.

The symbol δ ij refers to all of the components of the system simultaneously. As another example, consider
the equation
m, n =1, 2, 3 (1.1.1)
b e m · b e n = δ mn
the subscripts m, n occur unrepeated on the left side of the equation and hence must also occur on the right
hand side of the equation. These indices are called “free ”indices and can take on any of the values 1, 2or3
as specified by the range. Since there are three choices for the value for m and three choices for a value of
n we find that equation (1.1.1) represents nine equations simultaneously. These nine equations are

b e 1 · b e 1 =1 b e 1 · b e 2 =0 b e 1 · b e 3 =0
b e 2 · b e 1 =0 b e 2 · b e 2 =1 b e 2 · b e 3 =0

b e 3 · b e 1 =0 b e 3 · b e 2 =0 b e 3 · b e 3 =1.

Symmetric and Skew-Symmetric Systems

A system defined by subscripts and superscripts ranging over a set of values is said to be symmetric
in two of its indices if the components are unchanged when the indices are interchanged. For example, the
third order system T ijk is symmetric in the indices i and k if

T ijk = T kji for all values of i, j and k.

A system defined by subscripts and superscripts is said to be skew-symmetric in two of its indices if the
components change sign when the indices are interchanged. For example, the fourth order system T ijkl is
skew-symmetric in the indices i and l if

for all values of ijk and l.
T ijkl = −T ljki

As another example, consider the third order system a prs , p,r,s =1, 2, 3 which is completely skew-
symmetric in all of its indices. We would then have

a prs = −a psr = a spr = −a srp = a rsp = −a rps .

It is left as an exercise to show this completely skew- symmetric systems has 27 elements, 21 of which are
zero. The 6 nonzero elements are all related to one another thru the above equations when (p, r, s)= (1, 2, 3).
This is expressed as saying that the above system has only one independent component.