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+%%
+%% This is file `mathfont_example.tex',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% mathfont.dtx (with options: `example')
+%%
+%% Copyright 2018 by Conrad Kosowsky
+%%
+%% This file may be distributed and modified under the terms
+%% of the LaTeX Public Project License, version 1.3c or any later version.
+%% The most recent version of this license is available online at
+%%
+%% https://www.latex-project.org/lppl/.
+%%
+%% This work has the LPPL status "maintained," and the current maintainer
+%% is the package author, Conrad Kosowsky. He can be reached at
+%% kosowsky.latex@gmail.com. The work consists of the file mathfont.dtx,
+%% the derived files mathfont.sty, mathfont_greek.tex, and
+%% mathfont_example.tex, and all other files created through the configuration
+%% process such as mathfont.pdf, mathfont.idx, and mathfont.ind. For more
+%% information, see the original mathfont.dtx file.
+%%
+\documentclass[12pt]{article}
+\usepackage[margin=72.27pt]{geometry}
+\usepackage[factor=600,stretch=14,shrink=14,step=1]{microtype}
+\usepackage{selnolig}
+\usepackage[no-operators]{mathfont}
+\mathfont{Times New Roman}
+\mathfont[bb]{Symbola}
+\restoremathinternals
+\setmainfont{Times New Roman}
+\nolig{Th}{T|h}
+\hyphenpenalty=10
+\exhyphenpenalty=5
+\pretolerance=30
+\finalhyphendemerits=300
+\pagestyle{empty}
+\begin{document}
+
+\centerline{The \textsf{mathfont} Package in Action: Two Mathematical Snippets Rendered in Times New Roman}
+\centerline{Conrad Kosowsky}
+
+\bigskip
+
+Mathematicians usually define $e$ in one of two ways: as the horizontal asymptote of a certain function or as the limit of an infinite series. Specifically, it's most common to see $e$ defined as either
+\[
+e=\lim_{x\to\infty}\left(1+\frac1x\right)^x
+\]
+or
+\[
+e=\sum_{k=0}^\infty\frac1{k!}.
+\]
+The first definition is simpler in that involves a limit of a single expression, not a limit of partial sums, but in practice, the second tends to be more tractable. The power series expression of $e^x$ is given by
+\[
+\sum_{n=0}^\infty\frac{x^n}{n!},
+\]
+and the relationship between this expression and the series definition is much more apparent than it is for the first limit. This relationship arises in a variety of different mathematical contexts, for example the famous Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ or the related definition of the characteristic function for a random variable $X$:
+\[
+\phi_X(t)=\mathbb E\left(e^{iX}\right).
+\]
+Expanding $e^{iX}$ as a power series gives an expression for $\phi_X$ that we can differentiate term by term.
+
+\vfil
+
+A smooth manifold consists of a topological space $M$ equipped with a smooth maximal atlas $\leftbrace \phi_i\rightbrace$. The maps $\phi_i\colon U_i\longrightarrow\mathbb R$ technically aren't themselves differentiable, but their compositions $\phi_i^{}\circ\phi_j^{-1}$ are diffeomorphisms on subsets of $\mathbb R^n$. If we have a map $f\colon M\longrightarrow N$ between manifolds, this structure allows us to talk about differentiability of $f$. Specifically, we say that $f$ is smooth if for any $i$ and $j$, the composition
+\[
+\psi_j^{}\circ f\circ\phi_i^{-1}
+\]
+is itself smooth, where $\leftbrace\psi_i\rightbrace$ is a smooth atlas for $N$. Differentiating $f$ produces the associated tangent map $Df$. The function $Df$ maps the tangent space $TM$ to the tangent space $TN$ and is linear when restricted to individual tangent spaces $T_pM$. If $M$ can be written as a product $M_1\times M_2$, we can consider the partial tangent maps $\partial_1f$ and $\partial_2f$ by considering the compositions $f\circ\iota_1$ and $f\circ\iota_2$, where $\iota_1$ and $\iota_2$ are inclusion maps with respect to a particular point. Combining both maps, we have the equation
+\[
+Df(u,v)=\partial_1f(u)+\partial_2f(v),
+\]
+and this relationship can be thought of as an adaption of the standard product rule.
+
+\end{document}
+\endinput
+%%
+%% End of file `mathfont_example.tex'.