question about Einstein summation

Question

why is this expression valid?

$ c_{i}(x_{i}+y_{i}) $

it has the index repeated 3 times instead of 2 times.

Answer 1

Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. There are essentially three rules of Einstein summation notation, namely:

1. Repeated indices are implicitly summed over.

2. Each index can appear at most twice in any term.

3. Each term must contain identical non-repeated indices. The first item on the above list can be employed to greatly simplify and shorten equations involving tensors.

As per your question, notice that rule (2) above states that Each index can appear at most twice IN ANY TERM, so the index did not repeat 3 times. So it is valid.

Text rule source is from:wolframalpha.