# Problem with solution Simplify 6 Expressions with Kronecker Delta

Simplify the following expressions using the rules for calculating with Kronecker delta:

\(\delta_{31}\,\delta_{33}\)

\(\delta_{ji}\,T_{ink}\)

\(\delta_{j1}\,\delta_{ji}\,\delta_{2i}\)

\(\delta_{ik}\,\delta_{i3}\,\delta_{3k}\)

\(\delta_{jj}\) with \(j ~\in~ \{ 1,2,3,4 \} \)

\(\delta_{k\mu} \, \varepsilon_{kmn} \, \delta_{ss} \) with \(s ~\in~ \{ 1,2 \} \)

## Solution tips

Use the properties of Kronecker delta that you learned in the lesson. In all the following expressions you should keep in mind that \(\delta_{ik}\,\delta_{ij}\) can be summed up to \(\delta_{kj}\) and that you have to sum over equal indices:`$$ \delta_{ii} ~=~ 1~+~1~+~ ... ~+~ 1 ~=~ n $$`

## Solution for (a)

Here we simplify the following expression:`\begin{align}
\delta_{31}\,\delta_{33} ~&=~ 0 \cdot 1 ~=~ 0
\end{align}`

We exploited that \( \delta_{31} = 0 \) is because the indices have two different values and \( \delta_{33} = 1 \) has two equal indices.

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