Chapter 1, Problem 1.22P

(a)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $i=j$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(b)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $i=l$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(c)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $i=m$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(d)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $j=l$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(e)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $j=m$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(f)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $l=m$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(g)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $i\ne l\text{ or }m$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$

(h)

To determine

The sum ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}$ for $j\ne l\text{ or }m$ and show that ${\displaystyle \sum _k{\epsilon }_{ijk}{\epsilon }_{lmk}}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$. Also prove that $A\times \left(B\times C\right)=\left(A\cdot C\right)B-\left(A\cdot B\right)C$.