我正在尝试将方程式纳入我的 Beamer 演示文稿中，但是每次我这样做时，它都说它已因错误而退出。
我尝试过使用\begin{equation}和\end{equation}，也尝试过放置，$equation$但都没有用。我今天晚些时候要作为大学课程作业来展示这个演讲，所以真的需要帮助。谢谢。
\documentclass{beamer}
\usepackage{beamerthemesplit} % new
\usepackage[ps,dvips,all,color,line]{xy}
\usepackage[lite]{amsrefs}
\usepackage{graphicx}
\usepackage{verbatim}
\usepackage{amsmath,amssymb,enumerate,theorem,color,multirow,graphicx}
\usepackage[toc,page]{appendix}
\usepackage{fancyhdr}
\begin{document}
\title{A history of Ordinary differential equations and their applications within Biology, Physics and the economy}
\author{Tarran Dutton}
\date{\today}
\frame{\titlepage}
\begin{frame}[allowframebreaks]
\frametitle{Table of Contents}
\tableofcontents[sections=1-3]
\framebreak
\tableofcontents[sections=4-6]
\end{frame}
\section{1 - An introduction into Ordinary differential equations}
\frame{\frametitle{introduction to Ordinary differential equations}
\begin{itemize}
\item Within this presentation I will be taking a look into the history of Ordinary differential equations, and the people that led the way to establishing a fundamental piece of mathematics, that has uses across almost all platforms. Then will be looking into some of the practical applications that they have across Biology, Physics and the economy.
\end{itemize}
}
\frame{\frametitle{introduction to Ordinary differential equations}
\begin{itemize}
\item For around 350 years Ordinary differential equations have been commonplace mathematics across all sorts of fields, for example Newton’s famous F=ma is an ordinary differential equation. They have paved the way for countless other professions to analyse data and ultimately make the changes necessary to achieve their goals.
\end{itemize}
}
\frame{\frametitle{What is an Ordinary differential equation?}
\begin{itemize}
\item Firstly what is a Differential equation?
An differential equation is an equation involving derivatives of a function, for example y'(x)-y=0.
Furthermore, an ordinary differential equation is any differential equation which only has one variable.
\end{itemize}
}
\frame{\frametitle{introduction to Ordinary differential equations}
\begin{itemize}
\item Ordinary differential equations are used to find how the rate of change of one variable is affected by the differentials. They come in the form of a variable and derivatives of that variable in an equation. When solved they give a number of solutions as apposed to one like a standard algebraic equation.
\end{itemize}
}
\frame{\frametitle{introduction to Ordinary differential equations}
\begin{itemize}
\item So an example of a differential equation if we have y being a function of x could be
y’’+2y’=3y where y’ represents the derivative of y wrt x.
This can also be written in a few different ways which mean the same, eg;
f''(x)+2f'(x)=3f(x)
which means the same mathematically.
\end{itemize}
}
\section{2 - History of ODE's}
\subsection{- Origins of ODE's}
\frame{\frametitle{Origins of ODE's}
\begin{itemize}
\item ODE's have been used for many hundreds of years across the globe by mathematicians, as well as countless other people in all sorts of professions. Since their discovery they have become a more and more prominent method for obtaining accurate real life data as the years have gone by.
\item It was first, in 1675, that a Mr Gottfried Wilhelm put pen to paper touching on the ideas of differential equations, however this was a much more simplified and not complete version of the mathematical technique we know today.
\item Further along in 1679 was when Sir Isaac Newton got involved and really looked into these differential equations, and what he did was begin to classify them, which ultimately is where ordinary differential equations came to be.
\end{itemize}
}
\section{3 - Applications within Biology}
\subsection{- Population growth}
\frame{\frametitle{Applications within Biology}
\begin{itemize}
\item There are thousands upon thousands of applications within Biology for ordinary differential equations. However within this presentation i will only be focusing on two in particular, these being;
\begin{itemize}
\item Population growth
\item Disease spread
\end{itemize}
\end{itemize}
}
\frame{\frametitle{Population Growth}
\begin{itemize}
\item Population growth at its core is very much related to rate, which makes ordinary differential equations a prime candidate for mapping the information gathered when monitoring population growth of a certain species.
\item When mapping population growth there is a few factors that need considerin, these being;
\begin{itemize}
\item N - Representing the total population size
\item t - Representing the total time elapsed
\item E - Representing the initial population
\item r - Representing the rate of growth ( constant )
\end{itemize}
\end{itemize}
}
\frame{\frametitle{Population Growth}
\begin{itemize}
\item Population growth at its core is very much related to rate, which makes ordinary differential equations a prime candidate for mapping the information gathered when monitoring population growth of a certain species.
\item When mapping population growth there is a few factors that need considerin, these being;
\begin{itemize}
\item N - Representing the total population size
\item t - Representing the total time elapsed
\item E - Representing the initial population
\item $r$ - Representing the rate of growth ( constant )
\end{itemize}
\end{itemize}
}
\subsection{- Disease spread}
\frame{\frametitle{Disease spread}
\begin{enumerate}
\item Introduction to \LaTeX
\item Course 2
\item Termpapers and presentations with \LaTeX
\item Beamer class
\end{enumerate}
}
\frame{\frametitle{numbered lists with pause}
\begin{enumerate}
\item Introduction to \LaTeX \pause
\item Course 2 \pause
\item Termpapers and presentations with \LaTeX \pause
\item Beamer class
\end{enumerate}
}
\section{4 - Applications within Physics}
\subsection{- Nuclear Decay}
\frame{\frametitle{Nuclear Decay}
}
\subsection{- Atmospheric pressure}
\frame{\frametitle{Atmospheric pressure}
}
\section{5 - Applications within the Econonmy}
\subsection{- The Solow Swan Method}
\frame{\frametitle{The Solow Swan Method}
}
\subsection{- The Sethi Model}
\frame{\frametitle{The Sethi Model}
}
\section{6 - Conclusion}
\frame{\frametitle{Conclusion}
}
\end{document}
答案1
用于\begin{itemize}[<+->]覆盖。数学没问题：
\documentclass{beamer}
\usepackage{beamerthemesplit} % new
\usepackage{amsmath}
\begin{document}
\begin{frame}[allowframebreaks]{What is an Ordinary differential equation?}
\begin{itemize}[<+->]
\item Firstly what is a Differential equation?
An differential equation is an equation involving derivatives of a function,
for example $y'(x)-y=0$.
Furthermore, an ordinary differential equation is any differential equation
which only has one variable.
\item Ordinary differential equations are used to find how the rate of change
of one variable is affected by the differentials. They come in the form of a
variable and derivatives of that variable in an equation. When solved they give
a number of solutions as apposed to one like a standard algebraic equation.
\item So an example of a differential equation if we have y being a function of
$x$ could be
$y’’+2y’=3y$ where $y’$ represents the derivative of $y$ wrt $x$.
This can also be written in a few different ways which mean the same, eg;
\[ f''(x)+2f'(x)=3f(x) \]
which means the same mathematically.
\end{itemize}
\end{frame}
\end{document}