%% %% This is file `mathfont_example.tex', %% generated with the docstrip utility. %% %% The original source files were: %% %% mathfont.dtx (with options: `example') %% %% Copyright 2018-2019 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{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 $\phi_i\colon U_i\longrightarrow\mathbb R$ 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 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 %% %% End of file `mathfont_example.tex'.