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Level 3
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

Problem with solutionSimplify expressions with Kronecker delta

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}