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Problem with solution Simplify 6 Expressions with Kronecker Delta

Level 3 (with higher mathematics)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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

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

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

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

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

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

  6. \(\delta_{k\mu} \, \varepsilon_{kmn} \, \delta_{ss} \) 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.

Solution for (b)

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Solution for (c)

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Solution for (d)

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Solution for (e)

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Solution for (f)

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