\documentclass[12pt]{article}
\usepackage{hyperref,amsfonts,amsmath,amssymb}

\usepackage{graphicx}
%If you want to include images, then jpg png formats of images are recognized, for example with the command \includegraphics[height=2in]{filename}


%can add or comment out packages as desired

\author{Name} 
\title{Paper Template}

\begin{document}
\date{}
\maketitle





\begin{abstract}
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Vivamus ut quam vel ipsum porta congue ac sit amet urna. Fusce non purus sit amet sem placerat ultrices. Nulla facilisi. Morbi quis orci erat, a tincidunt sapien. Phasellus gravida tristique bibendum. Maecenas ac felis at felis lobortis cursus. 

Do not make the abstract  too long.  
\end{abstract}


%Overall, research, analyze and reflect on discipline-specific materials. The idea is to incorporate different levels and areas as appropriate to investigate mathematical knowledge from MAT 4140 in order to recognize the development of mathematical ideas from the past and the breadth of mathematics covered in MAT 4140, its impact on mathematical study, and its relation to the global mathematical community.




%Note, the table of contents only appears after TWO typesettings, like the reference citations.  The first is to find everything and the second put in the TOC.
\tableofcontents

\section{Introduction and Prior Experience}


First introduce the topic, which could include why you choose this topic.



Next discuss connections to other classes, if possible, and any prior experience with content related to the topic before the capstone, including how and in what context you covered related topics in classes and other experiences, including previous research experiences related to your curve, surface or metric form like the Pythagorean theorem and coordinates, as appropriate to the topic you chose.  
This section could include definitions and a summary of examples and important results. the review is both a reflection and contains mathematical examples that connect broadly to the topic and/or
other parts of the paper. 
{\bf This also serves as a general introduction to the topic aimed at others who may not have had the coursework you did.}

Most people will probably summarize some related content from Calculus with Analytic Geometry II  and Calculus with Analytic Geometry III and perhaps from other classes like linear algebra, physics courses, or others.

\subsection{MAT 1120: Calculus With Analytic Geometry}

\subsection{MAT 2130: Calculus With Analytic Geometry III}

\subsection{MAT 2240: Introduction to Linear Algebra}

\[
A=
\left(\begin{array}{ccc}
\cos{\theta} & -\sin{\theta}& 0\\
\sin{\theta}& \cos{\theta}& 0\\
0& 0& 1\\
\end{array}\right)
\]  represents a counterclockwise rotation by $\theta$ in the $x-y$ plane with the $z$ coordinate fixed.





\section{Historical Connections and Applications}



Look for historical significance and history of when the related people discovered or investigated your topic (include full names as well as dates), including at least one mathematician from a country outside of the US when possible (it could be someone who laid groundwork on the surface, or peripheral but connected work). If your surface or metric is named after someone, you should look up that person too.   
Overall, this should include diverse people and cultures who contributed as you look at the development of mathematical ideas from the past.  



Here are in text citations for the bibliography \cite{Cullen72, Kastrup08, Kleiner07, Mactutor, Rosenfeld88} which should be placed where appropriate, like \cite{Cullen72}.


This section should also highlight some historical and/or modern applications.


\section{Differential Geometry of ***}

This section should include differential geometry of the topic in your own words such as visualization, geometric intuition, and geometric analysis or other considerations. Be sure to use professional formatting.


You can wait to work on this section until you complete the relevant project in 4140.  Unless you have prior experience with differential geometry, the topic you choose will connect to assignments in 4140 so that the capstone project and work in 4140 will complement each other. Thus, much of the work for the project will be done in concert with MAT 4140 and the focus is on preparing the formal project.  If you selected another topic that isn't related to the 4140 projects, be sure to include differential geometry related to it. 


\section{Conclusion}

This section should conclude in some way.  Try to be creative.  For example, if you have ideas for what you might have liked to look at if you had more time, summarizing possible future directions can be interesting in a conclusion. Or you might look back as you make additional connections. 

The overall idea is to incorporate different levels and areas as appropriate to investigate mathematical knowledge from MAT 4140 in order to recognize the development of mathematical ideas from the past and the breadth of mathematics covered in MAT 4140, its impact on mathematical study, and its relation to the global mathematical community.  Consider adding to prior sections or to this section as needed to strengthen these connections. 

In addition, as you look back, consider creativity and personalization, such as why did you choose the topic, your own motivation and exploration. Reflection could include what you found most interesting, surprising,
enjoyable, or learned about yourself and differential geometry through the project. Personalization could be in the introduction and conclusion or spread throughout.

\section{Acknowledgements and References}

\section*{Acknowledgements}

I wish to acknowledge the importance of... or thank...


%-------------------------------------------
%        Begin References
%-------------------------------------------

\begin{thebibliography}{99}


\bibitem{Cullen72} Cullen, Charles.  \textit{Matrices and Linear Transformations, 2nd edition}. Reading, MA: Addison-Wesley Publishing Company, 1972.

\bibitem{Kastrup08} Kastrup, Hans A.  \textit{On the Advancements of Conformal Transformations and their Associated Symmetries in Geometry and Theoretical Physics.} Annalen der Physik (v. 17/9-10, 2008).

\bibitem{Kleiner07} Kleiner, Israel.   \textit{A History of Abstract Algebra.}  Boston:  Birkhauser, 2007.

\bibitem{Mactutor} O'Connor, John and Edmund Robertson. ``MacTutor History of Mathematics Archive: Mathematics of Transformations.'' Available online: 
\url{https://mathshistory.st-andrews.ac.uk/Darcy/transformation/} (accessed January 2022).

\bibitem{Rosenfeld88} Rosenfeld, Boris Abramovich. \emph{A History of Non-Euclidean Geometry:  Evolution of the Concept of a Geometric Space.}  New York:  Springer, 1988.

\end{thebibliography}



\vspace*{.5cm}
{\sl Author's e-mail address:}  emailaddress@appstate.edu
\end{document}               % End of document.


