function Tony_1304271_hw1 %insert the exact file name (as explained in the submission
% instructions) after "function"
%-
% ENGINEERING MATHEMATICS III - MATH 2Z03
% ASSIGNMENT #1
% FORWARD EULER METHOD FOR NUMERICAL SOLUTION OF ODEs
%-
%-
% Covers:
% -
function Name_XXXXXXX_hwN % where Name_XXXXXXX_hwN is the name of the file you submit without the extension .m
%-
% ENGINEERING MATHEMATICS IV - MATH 2ZZ3
% ASSIGNMENT #2:
% VECTOR FUNCTIONS, CURVES AND SURFACES
%-
%-
% Covers:
% - "Numerical Mathematics"
clc; close all; clear all;
% Matlab Assignment 3
%Question 1 - Plotting in 3D
u = linspace(0,2*pi,35);
v = linspace(0,2*pi,35);
[u,v] = meshgrid(u,v);
x = (4+3*cos(v).*cos(u);
y = (4+3*cos(v).*sin(u);
z = 6*sin(v);
%Plotting details
surf(x,y,z);
%(Optiona
LAST (family) NAME:
Test # 1
FIRST (given) NAME:
Math 2MM3
ID # :
February 12, 2009
Tutorial # :
Instructors:
Dr. J.-P. Gabardo
Dr. Z. V. Kovarik
Dr. R. Yapalparvi
Test duration: 1 hour
Instructions: You must use permanent ink. Tests submitted in pencil w
Math 2ZZ3 Tutorial Separation of Variables (again)
Michael Birch
March 25, 2014
Recall from class Separation of Variables (again)
To solve linear partial dierential equations (PDEs), it is often possible to simplify the
problem to a set of ordinary dieren
MATH 2ZZ3 WINTER 2014: SOLUTIONS TO SOME
PROBLEMS FROM THE NINTH SET
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 12.3.36
Expand the function f (x) = x on the interval 0 < x < in a Fourier
series.
Unlike previous problems asking for
MATH 2ZZ3 WINTER 2014: SOLUTIONS TO SOME
PROBLEMS FROM THE TENTH SET
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 13.1.15
Find product solutions to uxx + uyy = u by separation of variables.
We set u(x, y) = f (x)g(y) and substitute i
MATH 2ZZ3 WINTER 2014: SOLUTIONS TO SOME
PROBLEMS FROM THE EIGHTH SET
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 12.1.12
Show that
1, cos x/p, cos 2x/p, cos 3x/p, . . .
sin x/p, sin 2x/p, sin 3x/p, . . .
form an orthogonal system o
Math 2ZZ3 Tutorial Solving the Heat Equation, Wave
Equation and Laplaces Equation
Michael Birch
April 1, 2014
Note about notation
I will write
u
x
= ux (L, t)
x=L
for the purposes of boundary conditions since the RHS is a cleaner notation.
Recall from cla
Math 2ZZ3 Tutorial Space Curves
Michael Birch
January 14, 2014
Notes about notation
A
Even though I may not always type up my notes in L TEX, I will try to be consistent with
my notation, which is dened below.
I will denote a vector using an arrow over t
Math 2ZZ3 Tutorial Complex Fourier Series and
Separation of Variables
Michael Birch
March 18, 2014
Recall from class Fourier Series for Odd and Even Functions
(again)
Consider a function dened on an interval (p, p). In the special case that f (x) is an od
MATH 2ZZ3 WINTER 2014: SOLUTIONS TO SOME
PROBLEMS FROM THE ELEVENTH SET
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 13.3.1
Solve the heat equation ut = kuxx for 0 x L and t 0, subject to
the conditions u(0, t) = u(L, t) = 0 for all
Math 2ZZ3 Tutorial Tangents, Normals, Directional
Derivatives
Michael Birch
January 21, 2014
Recall from class Unit tangent and curvature
We dene the unit tangent as
T=
r (t)
dr
=
,
ds
r (t)
(1)
where r(s) is the arclength parametrization of the path r(t)
MATH 2ZZ3 WINTER 2014: SOME SOLUTIONS FOR
THE FOURTH SET OF PROBLEMS
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem (9.7.21). Verify
usual, and a is constant).
(a r) = 0 (where r = xi + yj + zk as
Let a = a1 i + a2 j + a3 k. Then
a r
MATH 2ZZ3 WINTER 2014: SOME PROBLEMS FROM
THE FIRST SET
Problem (9.1.3). Plot the curve r(t) = (t, 2t, cos t) for t 0.
If x = t, y = 2t, z = cos t, then notice that y = 2x; therefore the
entire curve lies in the plane y = 2x. This plane intersects the xy
MATH 2ZZ3 WINTER 2014: SOME SOLUTIONS FOR
THE THIRD SET OF PROBLEMS
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem (9.6.5). Sketch the level curve of the function f = x2 /4 +
y 2 /9 passing through the point (2, 3), and draw the gradie
Math 2ZZ3 Tutorial Polar Coordinates and Greens
Theorem
Michael Birch
February 11, 2014
Recall from class Double Integrals and Centre of Mass
We can integrate a function f (x, y) over a region R. We interpret this quantity as the
(signed) volume of the so
Math 2ZZ3 Tutorial Line integrals and double integrals
Michael Birch
February 04, 2014
Recall from class Curl and Divergence (again)
For a vector eld, F(r) = Fx (r) + Fy (r) + Fz (r)k, the curl is dened to be
i
j
i
curl F =
F=
j
k
x
y
z
.
(1)
Fx Fy Fz
It
Math 2ZZ3 Tutorial Tangent Planes, Divergence, Curl
Michael Birch
January 28, 2014
Example 1: 9.5.40 Gradient Decent Method
I talked previously about the gradient vector being used in a (simple) numerical optimization
scheme to nd extrema of multi-variabl
MATH 2ZZ3 WINTER 2014: SOME SOLUTIONS FOR
THE SECOND SET OF PROBLEMS
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem (9.3.3). Find the unit tangent T, principal normal N, binormal B, and curvature of the circular helix dened by r(t) =
(
Math 2ZZ3 Tutorial Spherical Coordinates,
Cylindrical Coordinates, the Divergence Theorem, and
Change of Variables
Michael Birch
March 4, 2014
Recall from class Spherical and Cylindrical Coordinates
Spherical coordinates (Fig. 1a) are related to Cartesian
Math 2ZZ3 Tutorial Surface Integrals, Stokes
Theorem and Triple Integrals
Michael Birch
February 25, 2014
Recall from class Greens Theorem (again)
Theorem: Let R be a simply connected region where its boundary R is a piecewise smooth
curve. Let P , Q, P/y
Math 2ZZ3 Tutorial Fourier Series
Michael Birch
March 11, 2014
Recall from class Orthogonal Functions
In linear algebra we say that two vectors are orthogonal if their inner product is zero (this
is a generalization of two vectors in R2 or R3 being at 90
MATH 2ZZ3 WINTER 2014: SOME SOLUTIONS FOR
THE SIXTH SET OF PROBLEMS
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 9.12.27
Find R x2 dA where R is the region bounded by the ellipse x2 /9 +
y 2 /4 = 1 using Greens theorem.
Let C be the
MATH 2ZZ3 WINTER 2014: SOME SOLUTIONS FOR
THE FIFTH SET OF PROBLEMS
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem (9.10.39). Evaluate
1
0
1
(1
x
+ y 4 )1 dydx
It is a little dicult to perform the inner integration by nding the
indenit
MATH 2ZZ3 WINTER 2014: SOLUTIONS TO SOME
PROBLEMS FROM THE SEVENTH SET
Questions/corrections/comments: patel222@math.mcmaster.ca.
Problem 9.15.55
Find the centroid of the solid bounded by the hemisphere z =
and the plane z = 0.
a2 x 2 y 2
The given solid
Math 2ZZ3: Engineering Mathematics IV
Course Outline - Summer 2014
The course web page can be found on Avenue to Learn
Instructor:
Dr. David Lozinski HH/315, x23409, lozinski@math.mcmaster.ca
Oce hours will be from 5pm to 6pm on the evenings of the lectur