Index Notation

In a term, if an index occurs twice, on a first symbol and then another, then there is a summation involving those two symbols.

When you read about this, most likely they will go a step further and say that the index has to be “up” on one and “down” on the other. You will see this in the example below where we show the Kronecker Delta \delta^j_i:

\delta^j_i v_j = v_i

One thing about this, if an index is “up” in the numerator and “up” in the denominator, that causes summation since being “up” in the denominator is the same as being “down” in the numerator. Because of this, summation will occur for the example shown below:

\dfrac {\partial F} {\partial a^i} \dfrac{\partial a^i} {\partial \mu}

Appendix A

In a shocking find, an article with a title like “A Primer on Index Notation” went through twelve pages of interesting explanations–and not once was there ever an index placed in the superscript position. Everything was subscript and as we understand it, that should be called Suffix Notation.

electromagnetic field tensor. You could indicate it by F. But then you could forget that it is a second order covariant tensor, or a 2-form. This can be indicated by F = F_{ab}.

Reference: https://www.physicsforums.com/threads/question-abstract-index-notation.271734/