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5

I think you have asked several questions. The first is about $x, y$ versus $x$, $y$. I think the second one is semantically and hence typographically right since the comma is not part of the mathematical expression. Your second example is a little ambiguous. There I would include the comma in the mathematics. An implicit question is the choice between ...

4

First things first: An algorithm most probabilly will never cover all use cases at once. A good rule of thumb is to assume that on average a LaTeX document will reach 90% of the final quality without any direct intervention. That said, inserting a \linebreak here and there should not be the end of the world, i.e. \begin{defn} The double integral of $f$ on ...

5

It is instructive to examine the exact definitions of $$and$$ that are provided by the LaTeX kernel (contained in the file latex.ltx, version early 2016): \DeclareRobustCommand$${% \relax\ifmmode\@badmath\else\fi}% \DeclareRobustCommand$${% \relax\ifmmode\ifinner$\else\@badmath\fi\else \@badmath\fi}% The main thing to note is that $$and$$ act as ... 0 \documentclass{article} \usepackage{amsmath} \usepackage{amsthm} \usepackage{mdframed} \theoremstyle{definition} \newtheorem{defn}{Definition} \surroundwithmdframed[leftmargin=-10pt,rightmargin=20pt]{defn} \begin{document} \begin{defn} The double integral of$f$on$R$, denoted as$\iint_R f(x,y) \, \operatorname{d}\!A\$ \end{defn} \end{document} Same code ...

3

According to ISO 80000-2:2009, Quantities and units---part2: Mathematical signs and symbols to be used in the natural sciences and technology, the upright i is the correct choice. Quantities which are not variable across time or context (such as immutable constants of nature) are upright while variables, contextual constants, running numbers (dummies), are ...

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