91Ó°ÊÓ

The first rank tensor is just a vector. A tensor of second rank has nine components (in three dimensions) in every rectangular coordinate system.

In part (a), there are no free indices.

$\begin{array}{c}i=j\\ =k\\ =m\end{array}$

Hence, the required value becomes as follows.

$\begin{array}{c}{\mathrm{δ}}_{ij}{\mathrm{δ}}_{jk}{\mathrm{δ}}_{km}{\mathrm{δ}}_{mi}=\underset{i=1}{\overset{3}{∑}}{\mathrm{δ}}_{ii}{\mathrm{δ}}_{ii}{\mathrm{δ}}_{ii}{\mathrm{δ}}_{ii}\\ =\underset{i=1}{\overset{3}{∑}}1\\ =3\end{array}$

The value is not zero if only if $j=k$

For $j=k$, the value of ${\mathrm{Î\mu }}_{ijk}$becomes as follows.

Hence, ${\mathrm{δ}}_{ij}{\mathrm{Î\mu }}_{ijk}=0$

$\begin{array}{c}i=j\\ =k\\ =m\end{array}$

The non-zero parts in the sum is$(j,k)=(1,3)or(3,1)$.

Hence, the required value becomes as follows.

$\begin{array}{c}{\mathrm{Ï\mu }}_{jk2}{\mathrm{Ï\mu }}_{j2k}={\mathrm{Ï\mu }}_{132}{\mathrm{Ï\mu }}_{321}+{\mathrm{Ï\mu }}_{312}{\mathrm{Ï\mu }}_{123}\\ =(−1)â‹\dots (−1)+1â‹\dots 1\\ =2\end{array}$

The required value is mentioned below.

The required value is mentioned below.

The required value is mentioned below.

$\begin{array}{c}{\mathrm{Ï\mu }}_{k31}{\mathrm{Ï\mu }}_{3k1}={\mathrm{Ï\mu }}_{131}{\mathrm{Ï\mu }}_{311}+{\mathrm{Ï\mu }}_{231}{\mathrm{Ï\mu }}_{321}+{\mathrm{Ï\mu }}_{331}{\mathrm{Ï\mu }}_{331}\\ =0+1â‹\dots (−1)+0\\ =−1\end{array}$

Hence, the required values are

$\begin{array}{l}\left(a\right){\mathrm{δ}}_{ij}{\mathrm{δ}}_{jk}{\mathrm{δ}}_{km}{\mathrm{δ}}_{mi}=3\\ \left(b\right){\mathrm{δ}}_{ij}{\mathrm{Î\mu }}_{ijk}=0\\ \left(c\right){\mathrm{Ï\mu }}_{jk2}{\mathrm{Ï\mu }}_{j2k}=2\\ \left(d\right){\mathrm{Ï\mu }}_{3jk}{\mathrm{Ï\mu }}_{kj3}=−2\\ \left(e\right){\mathrm{Ï\mu }}_{23i}{\mathrm{Ï\mu }}_{2i3}=−1\\ \left(f\right){\mathrm{Ï\mu }}_{k31}{\mathrm{Ï\mu }}_{3k1}=−1\end{array}$