%\documentclass[10pt]{article}\newcommand{\pix}{bmp} % for dvi
\documentclass[10pt]{amsart}\newcommand{\pix}{jpg} % for pdf
%\documentclass[10pt]{article}\newcommand{\pix}{jpg} % for pdf

\usepackage{graphicx}
%\usepackage{amsfonts}
%\usepackage{amsmath}
%\usepackage{amscd}
%\usepackage{xy}
%\usepackage{amssymb}

%Definitions that I'll probably want in most papers:
\newcommand{\br}{\ensuremath{\mathbb{R}}} %The real numbers
\newcommand{\bz}{\ensuremath{\mathbb{Z}}} %The integers
\newcommand{\bn}{\ensuremath{\mathbb{N}}} %The natural numbers
\newcommand{\bc}{\ensuremath{\mathbb{C}}} %The complex numbers
\newcommand{\bq}{\ensuremath{\mathbb{Q}}} %The natural numbers
\newcommand{\id}{\ensuremath{\mathrm{id}}} %The identity morphism
\newcommand{\bm}{\ensuremath{\mathbb{M}}} %The m for matrix algebra
\newcommand{\diam}{\ensuremath{\mathrm{diam}}} %The diameter
\newcommand{\e}{\ensuremath{\mathrm{e}}} %The number e

                            
\title{ }%: }  
\author{Bruce Hughes}
%\date{Preliminary Version: 21 July 2004}      % Deleting this command produces today's date.




\begin{document}

%$\left( \begin{array}{ccccc}
%\cos^2\phi_y^1 - \cos^2\phi_x^1 & \cos^2\phi^1_z - \cos^2\phi_x^1 & 2\cos\phi_x^1 cos\phi_y^1 & 2\cos\phi_x^1\cos\phi_z^1 & 2\cos \phi^1_y \cos\phi_z^1 \\ \bigskip
%\cos^2\phi_y^2 - \cos^2\phi_x^2 & \cos^2\phi^2_z - \cos^2\phi_x^2 & 2\cos\phi_x^2 cos\phi_y^2 & 2\cos\phi_x^2\cos\phi_z^2 & 2\cos \phi^2_y \cos\phi_z^2 \end{array} \right)
\begin{equation}

\begin{array}{ccccc}
\cos^2\phi_y^1 - \cos^2\phi_x^1 & \cos^2\phi^1_z - \cos^2\phi_x^1 & 2\cos\phi_x^1 cos\phi_y^1 & 2\cos\phi_x^1\cos\phi_z^1 & 2\cos \phi^1_y \cos\phi_z^1  \\
\cos^2\phi_y^2 - \cos^2\phi_x^2 & \cos^2\phi^2_z - \cos^2\phi_x^2 & 2\cos\phi_x^2 cos\phi_y^2 & 2\cos\phi_x^2\cos\phi_z^2 & 2\cos \phi^2_y \cos\phi_z^2 \end{array} \right)
\end{array}

\end{equation}



$D_i = D(v_i) \equiv k v_i^T S v_i$

\end{document}