Page 89 - Intro to Tensor Calculus

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Figure 1.3-14. Translation followed by rotation of axes

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In a similar manner one can show e ijk is a relative tensor of weight +1 and ijk = √ e ijk is an absolute
g
tensor. This is left as an exercise.

Another exercise found at the end of this section is to show that a generalization of the e − δ identity
is the epsilon identity
ij
g ipt jrs = g pr g ts − g ps g tr . (1.3.36)

Cartesian Tensors

Consider the motion of a rigid rod in two dimensions. No matter how complicated the movement of
the rod is we can describe the motion as a translation followed by a rotation. Consider the rigid rod AB
illustrated in the figure 1.3-13.

Figure 1.3-13. Motion of rigid rod

In this figure there is a before and after picture of the rod’s position. By moving the point B to B we
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have a translation. This is then followed by a rotation holding B fixed.