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Problem with solution Simplify expressions with Kronecker delta

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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}