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%% This is file `mathfont_example.tex',
%% generated with the docstrip utility.
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%% The original source files were:
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%% mathfont.dtx  (with options: `example')
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%% Copyright 2018-2019 by Conrad Kosowsky
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%% 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
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%%          https://www.latex-project.org/lppl/.
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%% 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.
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\documentclass[12pt]{article}
\usepackage[margin=72.27pt]{geometry}
\usepackage[factor=600,stretch=14,shrink=14,step=1]{microtype}
\usepackage{mathfont}
\setfont{Times New Roman}
\mathfont[bb]{Symbola}
\restoremathinternals
\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 $\phi_i\colon U_i\longrightarrow\mathbb R$ are continuous, and 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 bundle $TM$ to the tangent bundle $TN$ 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
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%% End of file `mathfont_example.tex'.
