# Preface

This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using R, but we strongly recommend to use RStudio. If you have never used R before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043.

This tutorial contains many R scripts. You, as the user, are free to use all codes for your needs, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. Any comments and/or contributions for this tutorial are welcome; you can send your remarks to <Vladimir_Dobrushkin@brown.edu>.

Introduction to Linear Algebra with R

# 1. 3D plotting

R provides many useful instructions for the visualization of 3D data. The instructions provided include tools to plot wire-frame objects, 3D plots, curves, surfaces, etc. and can automatically generate contours, display volumetric data, interpolate shading colors and even display non-Matlab made images.

 Order Function plot3(x,y,z,'line shape') Space curve meshgrid(x,y) Mesh plot mesh surf shading flat countour3 meshc [x,y,z]=cylinder(r,n)

We show how to use plotting commands by examples.

# Example 0.1.1:

In the interval [0, 10*pi], draw the parametric curve:
x = sin(t), y = cos(t), z = t .

>> t=0:0.01:10*pi;
>> plot3(sin(t),cos(t),t)

# Example 0.1.2:

Plot a surface z=(x+y)^2

>> x=-3:0.1:3;
>> y=1:0.1:5;
>> [x,y]=meshgrid(x,y);
>> z=(x+y).^2;
>> plot3(x,y,z)

When using meshgrid,'' the coding should be changed into:

>> x=-3:0.1:3;
>> y=1:0.1:5;
>> [x,y]=meshgrid(x,y);
>> z=(x+y).^2;
>> mesh(x,y,z)

Or the solid filled:

>>x=-3:0.1:3;
>> y=1:0.1:5;
>> [x,y]=meshgrid(x,y);
>> z=(x+y).^2;
>> surf(x,y,z)

And with the flat surface:

>>

# Example 0.1.3:

Toroid surface:

a=5; % a and c are arbitrary constants
c=10;
[u,v]=meshgrid(0:10:360); % meshgrid produces a full grid represented by the output coordinate arrays u and v; u and v are time variables
x=(c+a*cosd(v)).*cosd(u); % x, y, and z are the parameterized equations for a toroid
y=(c+a*cosd(v)).*sind(u);
z=a*sind(v);
surfl(x,y,z) % surfl creates a surface plot with colormap-based lighting
axis equal; % scales the axis so the x, y, and z axis are equal

# Example 0.1.4: Helix

t = 0:pi/50:10*pi; %create the vector t. starting value of 0, each element increases pi/50 from the previous up through 10*pi
plot3(sin(t),cos(t),t) %3D plot, coordinate (x, y, t)
title('Helix') %creates plot title
xlabel('sin(t)') %creates x label
ylabel('cos(t)') %creates y label
zlabel('t') %creates t label
grid on %dashed grids

# Example 0.1.5:

% The Viviani's Curve is the intersection of sphere x^2 + y^2 + z^2 = 4*a^2
%and cylinder (x-a)^2 +y^2 =a^2
%This script uses parametric equations for the Viviani's Curve,
%and the cylinder and sphere whose intersection forms the curve.

% For the sphere:
a = 2; %the parameter 'a' from the equations, here a=2
phi = linspace(0,pi,40); %this defines the scope of phi, from 0 to pi, the '40' indicates 40 increments between the bounds
theta = linspace(0,2*pi,50); %this defines the scope of theta
[phi,theta] = meshgrid(phi,theta); %creates a grid of (phi,theta) ordered pairs
%the parametric equation for the sphere:
x=2*a*sin(phi).*cos(theta); %use a '.' before the operator, this is an array operation
y=2*a*sin(phi).*sin(theta); %the . indicates that this operation goes element by element
z=2*a*cos(phi); %note: always suppress data using a semicolon
%Plotting the sphere:
mhndl1=mesh(x, y, z); %'mhndl' plots and stores a handle of the meshgrid
set(mhndl1,... % formats 'mhndl'
'EdgeColor',[0,0,1]) %gives the surface color, [0,0,1] is a color triple, vary the combination to get different color
axis equal %controls axis equality, this is very important for ensuring the sphere looks perfectly round
axis on %controls whether axes are 'on' or 'off'

%For the cylinder:
t=linspace(0,2*pi,50); %defines the scope of 't'
z=linspace(-2*a,2*a,30); %defines the scope of 'z', the bounds for the top and bottom of the cylinder
[t,z]=meshgrid(t,z); %creates pairs of (t,z)

x=a+a*cos(t); %the parametrized cylinder
y=a*sin(t);
z = 1*z;

%Plotting the cylinder
hold on %prevents override of plot data, so both the cylinder and sphere can occupy the same plot
mhndl2=mesh(x, y, z); %notice 'mhndl' is numbered, mhndl2=mesh(x,y,z) plots the cylinder
set(mhndl2,...
'EdgeColor',[1,0,0]) %[1,0,0] gives the color red
view(50,20) %view(horizontal_rotation,vertical_elevation) sets the angle of view, both parameters used are measured in degress

%For the actual Vivani's Curve:
t=linspace(0,4*pi,200); %scope of the 't' values
x= a+a*cos(t); %parametrized Viviani's Curve
y= a*sin(t);
z = 2*a*sin(t/2);
%To plot the Viviani's Curve:
vhndl = line( x,y,z); %here 'vhndl' is used to plot the curve and record the handle
set(vhndl, ...
'Color', [0,0,0],... % here [0,0,0] gives the color black
'LineWidth',3) %line width of the line
view(50,20); %similar to before, this indicates 50degrees horizontal rotation, 20 degress vertical elevation