function lab2soln2ZZ3  % 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('  ')

syms xs ys;

Fxs = sin(ys)*cos(xs)+0.5;
Fys = sin(xs)*cos(ys)+0.2;
phis = sin(xs)*sin(ys)+0.5*xs+0.2*ys;

Fx = matlabFunction(Fxs);
Fy = matlabFunction(Fys);
phi = matlabFunction(phis);

k = [1 2 3];
l = [1 2 3];
[K,L] = meshgrid(k,l);

x = linspace(0,4,100);
y = x;
[X,Y] = meshgrid(x,y);

figure('Name','A conservative vector field and its potential function');
title(['F(x,y) = < sin y cos x + 1/2, sin x cos y + 1/5 >' char(10) ...
    'and its potential function \phi = sin x sin y + x/2 + y/5']);
hold on; grid on;
set(gca,'PlotBoxAspectRatio',[1 1 1]);
quiver(K, L, Fx(K,L), Fy(K,L));
contour( X, Y, phi(X,Y), sort(phi(K,L)) );

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


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

% (a)

phi = @(x,y,z) x.^2 + x.*y + x.*y.*z;

Answer1 = phi(1,2,1) - phi(0,0,0);

% (b)

syms x y z t;
Fxs = 2*x + y + y*z; 
Fys = x + x*z;
Fzs = x*y;
Fs = [Fxs Fys Fzs];
rxs = sin(pi*t/2);
rys = 2*t;
rzs = t^2;
rs = [rxs rys rzs];

Fdrs = subs(Fs,[x y z],rs)*diff(rs,t)';

Fdr = matlabFunction(Fdrs);

Answer2 = quadl(Fdr,0,1);


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



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