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diff --git a/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex new file mode 100644 index 00000000000..efb26fe250c --- /dev/null +++ b/Master/texmf-dist/doc/latex/mathfont/mathfont_example.tex @@ -0,0 +1,80 @@ +%% +%% 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'. |