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\documentclass[aspectratio=169]{beamer}
\usepackage{amsmath,unicode-math,physics,tensor,xeCJK,bookmark}
\useoutertheme{metropolis}
\useinnertheme{metropolis}
\usecolortheme{metropolis}
\usefonttheme{professionalfonts}
\setbeamerfont{title}{size=\Large, series=\bfseries}
\setbeamerfont{author}{size=\small}
\setbeamerfont{date}{size=\small}
\setbeamertemplate{footline}{\vspace*{0.3cm}}
\makeatletter
% https://tex.stackexchange.com/q/66519
\apptocmd{\beamer@@frametitle}{\only<1>{\bookmark[page=\the\c@page,level=3]{#1}}}{}{}
\unimathsetup{math-style=ISO, bold-style=ISO, mathrm=sym}
\setsansfont{FiraGO}[BoldFont=* SemiBold, Numbers=Monospaced]
{ \setmathfont { FiraMath-##1.otf } [ version = ##1 ] }
\cs_new:Npn \MultipleWeights #1
{
\seq_map_inline:Nn \c_@@_weight_seq
{ \group_begin: \mathversion {##1} #1 \group_end: }
}
\ExplSyntaxOff
\makeatother
\def\ii{\symrm{i}}
\def\pp{\symrm{\pi}}
\title{Fira Math}
\subtitle{Sans-serif font with Unicode math support}
\author{Xiangdong Zeng}
\date{2020/10/15\quad v0.3.4}
\begin{document}
\maketitle
\begin{frame}{Basic examples (I)}
\begin{itemize}
\item Covariant derivative:
\[
\nabla \symbf{X} = \tensor{X}{^\alpha_{;\beta}} \pdv{x^\alpha} \otimes \dd{x^\beta}
= \qty(\tensor{X}{^\alpha_{,\beta}} + \Gamma^{\alpha}_{\beta\gamma} \, X^\gamma) \,
\pdv{x^\alpha} \otimes \dd{x^\beta}
\]
\item Einstein's field equations:
\[ G_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]
\item Schwarzschild metric:
\[
c^2 \dd{\tau}^2 = \qty(1-\frac{r_{\mathrm{s}}}{r}) \, c^2 \dd{t}^2
- \qty(1-\frac{r_{\mathrm{s}}}{r})^{-1} \dd{r}^2
- r^2 \underbrace{\qty(\dd{\theta}^2 + \sin^2 \theta \dd{\varphi}^2)}_{\dd{\Omega}^2}
\]
\item Einstein--Hilbert action:
\[ S = \frac{1}{2\kappa} \int R \sqrt{-g} \dd[4]{x} \]
\end{itemize}
\end{frame}
\begin{frame}{Basic examples (II)}
\begin{itemize}
\item Case $n=1$
\small
\[
\int_0^{\frac{\pp}{2}}
\frac{\sqrt{\frac12 \sqrt{\frac{\ln^2\cos\theta}{\theta^2+\ln^2\cos\theta}} + \frac12}}%
{\fourthroot{\theta^2 + \ln^2\cos\theta}} \dd{\theta}
= \frac{\pp}{2\sqrt{\ln 2}}
\]
\item Generalization:
\small\vspace{1ex}
\[
\begin{cases}
\smash[t]{\displaystyle
R_n^- = \frac{2}{\pp} \int_0^{\pp/2} \qty(\theta^2+\ln^2\cos\theta)^{-2^{-n-1}}
\sqrt{\frac12+\frac12\sqrt{\frac12+\cdots+\frac12\sqrt{
\frac{\ln^2\cos\theta}{\theta^2+\ln^2\cos\theta}}}} \dd{\theta}
= (\ln 2)^{-2^{-n}}} \\[3ex]
\smash[b]{\displaystyle
R_n^+ = \frac{2}{\pp} \int_0^{\pp/2} \qty(\theta^2+\ln^2\cos\theta)^{2^{-n-1}}
\sqrt{\frac12+\frac12\sqrt{\frac12+\cdots+\frac12\sqrt{
\frac{\ln^2\cos\theta}{\theta^2+\ln^2\cos\theta}}}} \dd{\theta}
= (\ln 2)^{2^{-n}}}
\end{cases}
\]
\end{itemize}
\end{frame}
\begin{frame}{Using with CJK fonts}
\begin{itemize}
\item {\fontzhhans 【留数定理】全纯函数 $f$ 在若尔当曲线 $\gamma$ 上的积分为：}
\[
\oint_\gamma f(z) \dd{z}
= 2\pp\ii \sum_{k=1}^n \Res_{z=a_k} f(z).
\]
\item {\fontzhhant 【留數定理】全純函數 $f$ 在若爾當曲線 $\gamma$ 上的積分為：}
\[
\oint_\gamma f(z) \dd{z}
= 2\pp\ii \sum_{k=1}^n \Res_{z=a_k} f(z).
\]
\item {\fontja 【留数定理】ジョルダン曲線 $\gamma$ に沿う正則関数 $f$ の積分は、}
\[
\oint_\gamma f(z) \dd{z}
= 2\pp\ii \sum_{k=1}^n \Res_{z=a_k} f(z).
\]
\end{itemize}
\end{frame}
\begin{frame}{Multiple weights (preview)}
\centering
\everymath{\displaystyle}
\begin{tabular}{p{5cm}l}
\mathversion{Thin} $ \pdv{\alpha} \sin\alpha = \cos \alpha $ & \mathversion{Medium} $ \int \sin x \dd{x} = -\cos x + C_1 $ \\[12pt]
\mathversion{UltraLight} $ \pdv{\beta } \cos\beta = -\sin \beta $ & \mathversion{SemiBold} $ \int \cos y \dd{y} = \sin y + C_2 $ \\[12pt]
\mathversion{ExtraLight} $ \pdv{\gamma} \tan\gamma = \sec^2\gamma $ & \mathversion{Bold} $ \int \tan z \dd{z} = -\ln|\cos z \, | + C_3 $ \\[12pt]
\mathversion{Light} $ \pdv{\theta} \cot\theta = -\csc^2\theta $ & \mathversion{ExtraBold} $ \int \cot p \dd{p} = \ln|\sin p \, | + C_4 $ \\[12pt]
\mathversion{Book} $ \pdv{\phi } \sec\phi = \tan\phi \sec\phi $ & \mathversion{Heavy} $ \int \sec q \dd{q} = \ln|\sec q + \tan q \, | + C_5 $ \\[12pt]
\mathversion{Regular} $ \pdv{\zeta } \csc\zeta = -\cot\zeta\csc\zeta $ & \mathversion{Ultra} $ \int \csc r \dd{r} = -\ln|\csc r + \cot r \, | + C_6 $
\end{tabular}
\end{frame}
\end{document}
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