# Problem with solution Simplify expressions with Kronecker delta

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. $$\delta_{ji}\,T_{ink}$$
2. $$\delta_{j1}\,\delta_{ji}\,\delta_{2i}$$
3. $$\delta_{ik}\,\delta_{i3}\,\delta_{3k}$$
4. $$\delta_{jj}$$ mit $$j ~\in~ \{ 1,2,3,4 \}$$
Solution tips

Use the properties of Kronecker delta that you learned in the lesson.

## Solution

Solution

In all the following expressions you should keep in mind that $$\delta_{ik}\,\delta_{ij}$$ can be summed to $$\delta_{kj}$$ and that you have to sum over equal indices:$\delta_{ii} ~=~ 1~+~1~+~ ... ~+~ 1 ~=~ n$

1. Solution to (a): $\delta_{ji} \, T_{ink} ~=~ T_{jnk}$
2. Solution to (b): First, sum over $$j$$ and then over $$i$$; this will lead to the result zero according to the definition of Kronecker delta: \begin{align} \delta_{j1} \, \delta_{ji} \, \delta_{2i} &~=~ \delta_{1i} \, \delta_{2i} \\\\ &~=~ \delta_{12} \\\\ &~=~ 0 \end{align}
3. Solution to (c): Proceed analogously to (b), except that at the end you get not zero but one: \begin{align} \delta_{ik} \, \delta_{i3} \, \delta_{3k} &~=~ \delta_{k3} \, \delta_{3k} \\\\ &~=~ \delta_{33} \\\\ &~=~ 1 \end{align}
4. Solution to (d): Sum up over $$j$$ up to 4:\begin{align} \delta_{jj} &~=~ \delta_{11} ~+~ \delta_{22} ~+~ \delta_{33} ~+~ \delta_{44} \\\\ &~=~ 1~+~1~+~1~+~1 \\\\ &~=~ 4 \end{align}