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When writing a document I find that I am switching back and forth between indicial notation for spatial coordinates. I would like to get your thoughts on the following examples accompanied with questions.

For context, I am working in a right-handed Cartesian (x,y,z) coordinate system.

To begin , consider how I state one of my equations as follows:

$$\tag{1} S_{ij}x_ix_j=1$$ where where $x_i$ and $x_j$ ($i,j=1,2,3$ and $x_1\equiv x, x_2\equiv y, x_3 \equiv z$) denote the coordinates of a point that satisfy Equation (1).

So in the equation above, using the double-index summation convention, it seems natural to use the numbers 1,2,3 rather than the letters x,y,z, due to the $x_i$ and $x_j$ variables. If the letters x,y,z were used then one would have, as an example $(S_{ij}=S_{ji})$: $$\tag{2} S_{xx}x_x^2+S_{yy}x_y^2+...+2S_{xy}x_xx_y=1$$ Although this seems to me valid, I just have never seen it written this way. What is your take?

As a "subset" question, when defining the values of an index, should one write it as I have done? E.g., $i=1,2,3$? Or should it be stated as $i \in \text{{1,2,3}}$? The latter I have seen used, but not as much. Is one more technically correct?

For equation (1) I clarify that $x_1\equiv x, x_2\equiv y, x_3 \equiv z$ because I provide plots of the surface defined by Eqn(1), where the coordinate axes are labelled as "x", "y", "z", and this is the only option for the axes labels per my plotting software.

So perhaps I just write all my equations in a form that do not use numerical indicial notation? E.g., in vector-matrix form, where vectors and matricies (& tensors) are written in bold Roman: $$\tag{3} \text{x}^T\text{Sx}=1$$ where $$S = \left( \begin{array}{ccc} a & h & g \\ h & b & f \\ g & f & c \\ \end{array} \right); \ \ x=(x \ y \ z)$$

Should one try to stick with one "form" for writting equations or is it OK to use the various forms as long as one is clear what the notations signify? Is there a simple & concise way of doing the latter?

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    $\begingroup$ As long as you define what your symbols are, you can choose whatever you want. You can take a coordinate system with coordinates called Pete, Rupert and John. It does not matter $\endgroup$ – Mathphys meister Feb 21 at 17:11
  • $\begingroup$ Numerical indices are fine. If you want to use $xyz$ indices, don’t ever write $x_y$; just write $y$. For example, $2S_{xy}xy$. If you use matrix-vector notation, write the vector as a column and not a row. $\endgroup$ – G. Smith Feb 21 at 17:14
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In some sub-fields people have conventions for what the "right" thing is, but the following rules generally should guide you to what makes sense:

The zeroth rule: you are writing to communicate something accurately. This supersedes all other rules, whether of grammar or style. Think about what your reader will (mis)understand.

The first rule: be consistent. No matter what you choose, decide on one notation and keep it consistent throughout the text.

The second rule: do not introduce the risk of accidental confusion. If something can be confused by a reader, it should be changed.

In your example equation (1) and (2) you have a position variable named $x$ and a coordinate named $x$ - that will likely cause confusion. So if positions are called $x_\text{something}$ the somethings should not be called $x$ too. Or you should call position vectors something different than $x$.

Coordinate-free notation is often a nice solution, except when it makes it complicated to see what is going on. In some fields you can just assume everything is in Cartesian 3D, in others not.

(The third rule: readers will usually miss your clarifications. If something needs clarification, it should probably be re-expressed until it doesn't. Then add a clarification unless it violates rule 1 or 2.)

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    $\begingroup$ This applies to way more than just this particular question! $\endgroup$ – Jon Custer Feb 21 at 18:02

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