function lab1soln2ZZ3  % where Name_XXXXXXX_hwN is the name of the file you submit without the extension .m

%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS IV - MATH 2ZZ3
%%%                             ASSIGNMENT #5:
%%%       TRIPLE INTEGRALS, DIVERGENCE THEOREM AND CHANGE OF VARIABLES
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 7.7--7.8.
% "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright, section 9.15, 9.16, 9.17
%------------------------------------------------------------------------

%------------------------------------------------------------------------
% Instructions:
% - Submit your assignment electronically (via Email) to the
%   address specific to your last name as indicated on the course
%   website; hardcopy submissions will not be accepted.
% - It is obligatory to use the current MATLAB template file available at
%   http://www.math.mcmaster.ca/gabardo/MATH2ZZ3/frames/template.m; 
%   submissions non compliant with this template will not be accepted.
% - Make sure to enter your name and student I.D. number in the
%   appropriate section of the template.
% - Late submissions and submissions which do not comply with
%   these guidelines will not be accepted.
% - All graphs should contain suitable titles and legends.
%------------------------------------------------------------------------
% Written by Vladislav Bukshtynov, 2009


clc; close all; clear all;

% Student information
  
disp('  ------------   Student Information  ------------  ')  

% Please enter your information here
Name = 'Lab 1';
Surname = 'Solutions';
ID = 1234567;

fprintf('  Student:  %s %s (ID: %d) \n', Name, Surname, ID);



disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #1  ----------  ')  
disp('  ')

% (a)

syms T;
rxs = (2+3*cos(3*T)).*cos(2*T);
rys = (2+3*cos(3*T)).*sin(2*T);
rzs = sin(3*T);
rs = [rxs rys rzs];
Ts = (rxs^2+rys^2+rzs^2)^(-1/2)*diff(rs,T);
Txs = Ts(1);
Tys = Ts(2);
Tzs = Ts(3);
Ns = (Txs^2+Tys^2+Tzs^2)^(-1/2)*diff(Ts,T);
Nxs = Ns(1);
Nys = Ns(2);
Nzs = Ns(3);
Bs = cross(Ts,Ns);
Bxs = Bs(1);
Bys = Bs(2);
Bzs = Bs(3);

rx = matlabFunction(rxs);
ry = matlabFunction(rys);
rz = matlabFunction(rzs);
Tx = matlabFunction(Txs);
Ty = matlabFunction(Tys);
Tz = matlabFunction(Tzs);
Nx = matlabFunction(Nxs);
Ny = matlabFunction(Nys);
Nz = matlabFunction(Nzs);
Bx = matlabFunction(Bxs);
By = matlabFunction(Bys);
Bz = matlabFunction(Bzs);

t = linspace(0,2*pi,100);
tv = 0:pi/3:2*pi;

scrsz = get(0,'ScreenSize');
figure('Name','Unit tangent, principal normal, and binormal vectors', ...
       'OuterPosition',[1 scrsz(4)/3 2*scrsz(3)/3 2*scrsz(4)/3]);
title(['r(t) = (2 + cos 3t)cos 2t \bf{}i\rm{} + ' ...
       '(2 + cos 3t)sin 2t \bf{}j\rm{} + sin 3t \bf{}k\rm{}' ...
       char(10) 'and its tangent, normal, and binormal vectors']);
hold on; grid on; view(3);
plot3(rx(t),ry(t),rz(t));
quiver3(rx(tv),ry(tv),rz(tv),Tx(tv),Ty(tv),Tz(tv));
quiver3(rx(tv),ry(tv),rz(tv),Nx(tv),Ny(tv),Nz(tv));
quiver3(rx(tv),ry(tv),rz(tv),Bx(tv),By(tv),Bz(tv));
legend('\bf{}r\rm{}(t)','unit tangent','principal normal','binormal');

% (b)

t = linspace(0,2*pi,151);
rps = sqrt(diff(rxs,T)^2+diff(rys,T)^2+diff(rzs,T)^2);
rp = matlabFunction(rps);

s = zeros(1,151);
s(1) = 0;
for i = 2:151
    s(i) = s(i-1) + quadl(rp,t(i-1),t(i));
end

figure('Name','Arc length function');
title('plot of the arc length function for the curve \bf{}r\rm{}');
hold on; grid on;
plot(t,s);

disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #2  ----------  ')  
disp('  ')

x = linspace(-4,4,151);
y = x;
[X,Y] = meshgrid(x,y);

figure('Name','3D contour + surface plot');
title('graph of sin(6 \surd(x^2+y^2)+3x)/\surd(x^2+y^2)');
hold on; grid on; view(3);
surfc(X,Y,sin(6*sqrt(X.^2+Y.^2)+3*X)./sqrt(X.^2+Y.^2));

disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')






disp('  ----------            Answers         ----------  ')  
disp('  ')
% DO NOT TOUCH THIS PART (RESERVED FOR THE INSTRUCTOR'S USE)

Name 
Surname 
ID



Answer1
Answer2
Answer3
Answer4
Answer5
Answer6




% User's Functions Space is below --------------------------