Page 20 - Intro to Tensor Calculus

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since e ijk = e jki . We also observe from the expression

F i = e ijk C j A k

that we may obtain, by permuting the symbols, the equivalent expression

F j = e jki C k A i .

This allows us to write
~
~
~
~
~
~
~
~
A · (B × C)= B j F j = B · F = B · (C × A)
which was to be shown.
~
~
~
The quantity A · (B × C) is called a triple scalar product. The above index representation of the triple
scalar product implies that it can be represented as a determinant (See example 1.1-9). We can write

A 1 A 2 A 3

~ ~ ~

A · (B × C)= B 1 B 2 B 3 = e ijk A i B j C k

C 1 C 2 C 3
A physical interpretation that can be assigned to this triple scalar product is that its absolute value represents
~ ~ ~
the volume of the parallelepiped formed by the three noncoplaner vectors A, B, C. The absolute value is
needed because sometimes the triple scalar product is negative. This physical interpretation can be obtained
from an analysis of the figure 1.1-4.

Figure 1.1-4. Triple scalar product and volume