Evaluating Expressions with Kronecker Deltas

Sep 2018
1
0
USA
I am taking a class involving vector/tensors and stumbled on the following exercises:

1) Evaluate the expression \(\displaystyle \delta _i^j \delta _j^i\)
2) Evaluate the expression \(\displaystyle \delta _i^i \delta _j^j\)
3) Simplify the expression \(\displaystyle \delta _j^i S_i S^j\)

I know that you can contract expressions with Kronecker deltas if there are similar indices as they effectively "cancel out", sort of.
So, would the answer to the first one just be 0? If so, then how do I approach the second one?

Thanks in advance for any help.
 
Jan 2009
431
122
I would apply definition. https://en.wikipedia.org/wiki/Kronecker_delta
I don't know much about the domain of Kronecker Deltas, but gave it an old college try.

For example, for evaluating (1) I'd use the piecewise definition, and extend when taking it to the power j.


There are four cases now for our new piecewise definition.
When both i and j not equal to 0. Result: 0
When i not equal to 0, j equal to 0. Result: Undefined / 0^0
When i equal to 0, j not equal to 0. Result: 1
When both i and j equal to 0. Result: 0

Do the same for delta j to the power of i.
When both i and j not equal to 0. Result: 0
When i not equal to 0, j equal to 0. Result: 1
When i equal to 0, j not equal to 0. Result: Undefined
When both i and j equal to 0. Result: 0

Take the product of these two expressions and you'll get

When both i and j not equal to 0. Result: 0
When i not equal to 0, j equal to 0. Result: Undefined
When i equal to 0, j not equal to 0. Result: Undefined
When both i and j equal to 0. Result: 0
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
I am taking a class involving vector/tensors and stumbled on the following exercises:

1) Evaluate the expression \(\displaystyle \delta _i^j \delta _j^i\)
\(\displaystyle \delta_i^j\) is 1 when i= j, 0 otherwise. Further since we are not given values of i and j the summation convention is used. Assuming the dimension here is 3, then \(\displaystyle \delta_i^j\delta_j^j= \sum_{I= 1}^3 \sum_{j= 1}^3 \delta_i^j\delta_j^i= (\delta_1^1\delta_1^1+ \delta_2^2\delta_2^2+ \delta_3^3\delta_3^3= 1+ 1+ 1= 3$.

2) Evaluate the expression \(\displaystyle \delta _i^i \delta _j^j\)
3) Simplify the expression \(\displaystyle \delta _j^i S_i S^j\)

I know that you can contract expressions with Kronecker deltas if there are similar indices as they effectively "cancel out", sort of.
So, would the answer to the first one just be 0? If so, then how do I approach the second one?

Thanks in advance for any help.[/QUOTE]\)