# 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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