The linear operator notation is depreciated ( with possible exception of theoretical mechanics and other domains where index notation is traditionally avoided ).
22.
The tensor index notation applies to any object which has entities that form multidimensional arrays not everything with indices is a tensor by default.
23.
Therefore many calculi of explicit substitution avoid variable names altogether by using a so-called " name-free " De Bruijn index notation.
24.
The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates.
25.
The notations in this article are : lowercase bold for four dimensional vectors ( except for the four-gradient ), and tensor index notation.
26.
The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time.
27.
The index notation used in physics makes it far more convenient for practical calculations, although it makes the overall geometric structure of the theory more opaque.
28.
In abstract index notation, to contract two tensors with each other, one places them side by side ( juxtaposed ) as factors of the same term.
29.
Indeed, the expression can be considered to not be using any coordinate system at all, if you consider the expression to be using abstract index notation.
30.
This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.
