%Gradients %If you modify this file, please indicate here and in the footnote %Math263D \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Gradients \footnote{Copyright \copyright 2002 Steve Chapin, and Todd Young. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} \begin{enumerate} \item Enter the following commands:\\ \verb& [X, Y] = meshgrid(-2:.2:2); & \\ \verb& Z = X.*exp(-X.^2 - Y.^2);&\\ \verb& mesh(X, Y, Z)&\\ Rotate this plot into various positions until you fully understand the shape. \item Next enter the following:\\ \verb& [DX, DY] = gradient(Z);&\\ \verb& contour(X,Y,Z) &\\ \verb& hold on&\\ \verb& quiver(X,Y,DX,DY)&\\ \verb& hold off& \item Repeat the above steps for the functions: $f(x, y) = exp(-x^2 - y^2)$ and $f(x, y) = x + y$. \item What is the relationship between contour (level) curves and gradient vectors? \item Write a brief report, using complete sentences and standard mathematical notation. \end{enumerate} \vfill \noindent \textsf{ The goal of this project is to familiarize the user with the vector field plot capabilities of the program and aid in the understanding of the meaning of the gradient.} \end{document}