Confusion about subscript vs superscript statement

Question

The author of a book on Tensor algebra and analysis makes this statement.

"The sense of the index changes (from superscript to subscript or vice versa) if it appears under the fraction bar."

Can someone clarify this for me? What fraction bar?

Answer 1

I believe they are trying to explain the following convention:

Say you have an expression like $$a^j\frac{\partial}{\partial x^j}.$$ We have one upper index, plus an upper index in the denominator. We view the latter as being a lower index, so the amount of upper and lower indices still match, making the net index still zero. In particular, an upper index in the denominator is considered a lower index.