I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian Tensors.
I know the rules regarding free suffix's and dummy suffixes, but I'm having trouble proving the substitution rule: δi j δj k = δi k
Assuming a range of 3 for each component, I will choose that the free suffix's have the following values: i=3, k=1. Therefore, using the implicit summation rule for the dummy index: δi j δj k = δ3 1 δ1 1 + δ3 2 δ2 1 + δ3 3 δ3 1 = ?
Is my expansion correct?
Many thanks.
Yes, your expansion is correct. Now note that at least one of the deltas in each product is zero, so the sum is zero. If you set $i=k$ one of the products will be $\delta_{ii}\delta_{ii}=1$