Subscripted (or superscripted) variables in Mathematica

Question

When I first learned of Mathematica and started to use it, I soon discovered that Mathematica supported subscripted and superscripted variables such as $M_{i j}$. Naively, I first thought that I could refer to a matrix element merely by specifying it's row and column indices such as $me_{1 2}$ or whatever for a matrix named $me$.

But, that is not the case, the notation for such elements is using the Part function or the $[[row]][[col]]$ notation.

Further, as I started a self-study of General Relativity, I thought about using scripted (sub or super) variables in Mathematica to manipulate tensor equations. Not so easy and I thought it would be automatic. I have since downloaded a few tensor calculus packages and find that they did not make use of subscripts or superscripts.

Therefore, since what I few are the obvious uses of subscripting and superscripting variables in Mathematica are not being used, just why is this notation supported. Or, maybe I am missing something and there is a big area of usage I am not aware of.

I do admit that among the ilk of this forum I must consider myself still a beginner as a user of Mathematica.

Answer 1

Naively, I first thought that I could refer to a matrix element merely by specifying it's row and column indices such as $me_{12}$ or whatever for a matrix named $me$.

In fact you can do just that using the subcript form of Part.

SeedRandom[123];
me = RandomInteger[{1, 50}, {3, 3}];
MatrixForm@me

First note that you don't need me[[row]][[col]] as you can specify further dimensions in one Part call. me[[row, col]].

me[[2, 2]]

26

Next note that there is a subscript form for Part. Type the following:

me Ctrl+- Esc [[ Esc 2 , 2 Esc ]] Esc Ctrl+Space

Esc [[ Esc and Esc ]] Esc product special characters for Part. That makes your code easier to read.

Hope this helps.

Answer 2

As of Version 10.0, you can use Indexed.

For example, you could use

Indexed[g, {μ, ν}]

to generically denote the metric tensor. Then, if g has explicitly been set to an array,

g = {{-1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}

then Indexed will extract the appropriate Part:

Indexed[g, {1,1}]

-1

It is too bad that since Indexed is deeply tied to Part, you cannot use the computer scientist and/or general relativist's convention of starting indices from 0.

Answer 3

These notations exist to be defined by you (the user) and therefore making your life easier (or harder, depending on the excess of usage). Consider reading: Operators without Built‐in Meanings

The only important thing to note is that when you are working with sub- or superscripted variables, the actual assignment is made to the corresponding function. (Not to the symbol being scripted.)

So when you assign something to a_i, the actual assignment is made to Subscript[a, i]. That's why syntax highlighting won't indicate your symbol a as defined. (And this is why I recommend not to use sub or superscripted variables.)

Aside from that you can add multiple definitions to achieve taking part of a List and using partial derivative at the same time:

Subscript[a_List, i__] := Part[a, i]
Subscript[a_, i_] := D[a, i]

Note: you can always use FullForm to find out the actual name of the underlying symbol (for example FullForm[a^*] yields SuperStar[a], and Definition to see existing definitions.

Answer 4

I use subscripted variables extensively in analysis of summations etc (e.g. derivation of Cramer-Rao lower bounds for cases with large numbers of variables). As a very simple example, find the value of m that minimises the following sum.

sum = Sum[(Subscript[x, i] - m)^2, {i, 1, n}]
eqn1 = D[sum, m] == 0
eqn2 = eqn1 /. u_Sum :> (u /. Subscript[x, i] -> 0) + (u /. m -> 0)
sol = Solve[eqn2, m]

Unfortunately Mathematica is rather reluctant to make obvious simplifications (e.g. cancelling factors inside and outside the summation) so I have had to employ extensive sets of rules that I've developed myself.