Mathematics 264 Advanced Calculus Assignment 4
Due in class Thursday October 14, 2010
1. (a) Calculate the curl of F = rn (x i + y j), where r2 = x2 + y 2 , and n
is an integer, i.e. n = . . . , 2, 1, 0, 1, 2, . . .
(b) For each n for which curl F = 0, nd
Multiple Integrals
Advanced Calculus
Lecture 3
Dr. Lahcen Laayouni
Department of Mathematics and Statistics
McGill University
January 11, 2007
Dr. Lahcen Laayouni
Advanced Calculus
Multiple Integrals
Triple integrals Change of variables Cylindrical coordi
Vibrations of an Elastic String 635
)rtrlal solution of the problem of Eqs. (1), (3), and (9) is given by Eq. cfw_20)
.efcients calculated from Eq. (22).
ed value of n the expression sin(n7tx/L) eosnyrat/L) in Eq. (19) is peri
e I with the period 2L/na; i
Math 264 Notes from Class of Oct. 11, 2016
Summary of Lecture:
Normals Vs Unit Normals in the Computation of Flux
The Divergence and Curl of a Vector Field: Definition
The Divergence and Curl of a Vector Field: Physical Interpretation and Examples
Norm
Math 264 Notes from Class of November 15, 2016
Summary of Lecture:
Recap The Fourier Cosine Series and the Full Fourier Series. The Differences
Convergence: what do these Fourier Series give you?
What is special about sines and cosines?: Eigenvalue Pro
Math 264 Notes from Class of October 25, 2016
Summary of Lecture:
Finish Stokes question from last class
A summary of Stokes Theorem and a Comparison with the Other Fundamental Theorems.
Differentiation Under the Integral Sign
Incompressible Vector Fi
Math 264 Notes from Class of Nov. 24, 2016
Summary of Lecture:
Do ALL the material from the Notes of Nov 22 (we did not cover any of it!)
The Wave Equation
Different Boundary Conditions
Separation of Variables for the wave equation.
Summary of Separa
Math 264 Notes from Class of Sept. 20, 2016
Summary of Todays Lecture:
Finalize the date for the in class midterm no aides or calculators allowed on this
midterm.
Problem 8 on Webwork 1
Finish material from last class on an important 2D example (I repe
616
Chapter I 0. Partial Differential Equations and Fourier Series
EXAMPLE
1
that the innite series of Eq. (19) converges and also satises Eqs. (1) and (4). To
satisfy the initial condition (3), we must have
u(x,0) = E 2c" sin :x m f(x). (20)
12:1
In
Santander-Torres 1
Vannessa Santander-Torres
Andrew Willmer
345-LPH-MS: Ethics on Stage
May 6th 2016
Adapted to Program Assignment on C-section and Vaginal Birth
Babies can enter this world in one of two ways: Pregnant women can have either
a vaginal birt
McGill University
Faculty of Science
April 16, 2014
Final examination
Advanced Calculus for Engineers
Math 264
April 16, 2014
Time: 6PM-9PM
Examiner: Prof. R. Choksi
Associate Examiner: Prof. A. Hundemer
Student name (last, first)
Student number (McGill I
Math 264: Advanced Calculus
Winter 2007
FINAL EXAM
Solutions
Problem 1. Compute
Z
x+y+
C
y
2
x + y2
x
dx + y x + 2
x + y2
where C is the ellipse x2 /16 + 9y 2 = 1.
R
Solution. We have to compute C F ds, where F = x + y +
dy,
x
, y x + x2 +y
.
2
, x
. By
MATH 234 FINAL EXAM REVIEW PROBLEMS
WITH SOLUTIONS
You should be able to do all of the previous two review problem sets, plus the following problems.
Problem 1. Show that C 4x3 ydx + x4 dy = 0 for any closed curve C.
Solution. There are two ways to do thi
McGill University
Midterm examination (Solutions)
Nov. 3, 2016
Advanced Calculus for Engineers
Math 264
Nov. 3, 2016
Time: 14:35 - 16:00
Prof. R. Choksi
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS
1. You are allowed one double-sided
McGill University
Midterm examination (Revised Version)
Nov. 3, 2016
Advanced Calculus for Engineers
Math 264
Nov. 3, 2016
Time: 14:35 - 16:00
Prof. R. Choksi
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS
1. You are allowed one double
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R 14 Multiple Integration
Ie region D of Example 8
—o—o——~v—u-
1/2 1 M
Le transfomled region R
y=x
x2+4y2:4
1 2 x
rmajn D, Example 9
gion R, Example 9
1
Use an appropriate change of variables to ﬁnd the area of the elliptic
M disk E given by
2 2
x y
ﬁ
Math 264 Notes for Class of Dec 1, 2016 - Includes Solutions for PS2
PLEASE fill out the online Minerva course evaluations.
You have until Dec 6.
Review of the PDE/Fourier Series.
Review Advanced Calculus Component of Course by focusing on Exam Winter
Math 264 Notes from Class of Oct. 6, 2016 (as mentioned in class, I
have added a few extra problems
Summary of Lecture:
Finish Material on Flux Integrals with a few examples.
Divergence and Curl of a Vector Field.
1. A Few Think-First Examples of Flux I
Math 264 Notes from Class of Oct. 20, 2016
Summary of Lecture:
A Few Comments on the Divergence Theorem and Its Interpretations
The 2D Divergence Theorem
Stokes Theorem
You will be allowed one doubled-sided 8.5 x 11 inches sheet of information for
the
Math 264 Notes from Class of October 27, 2016
Summary of Lecture:
The Change of Variables Formula for Double Integrals (Two Variables): Examples
The Change of Variables Formula for Double Integrals (Three Variables)
Introduction to Partial Differential
Math 264 Notes from Class of Oct. 13, 2016
Summary of Lecture:
Finish the Curl of a Vector Field: Physical Interpretation and Examples
Vector Identities: Irrotational and Solenoidal Vector Fields
Start Greens Theorem
Note that I have extended the deadl
Math 264 Very Brief Notes for Nov 1, 2016
The Midterm consists of 5 questions each worth 10 marks. Some questions have 2 parts.
Remember
it is up to and including Stokes Theorem.
you are allowed one cheat sheet see previous announcement for details.
no
Notes for the First Math 264 Class on Sept 6, 2016
R. Choksi
Why this course?
From the perspective of an engineering student, I encourage
you to think about the following:
1 Calculus is essential for the modelling and analysis of any
complex system wherei
Double Integrals
Advanced Calculus
Lecture 2
Dr. Lahcen Laayouni
Department of Mathematics and Statistics
McGill University
January 9, 2007
Dr. Lahcen Laayouni
Advanced Calculus
Double Integrals
Polar coordinates
Change of variables formula
Double Integra
Multiple Integrals
Advanced Calculus
Lecture 1
Dr. Lahcen Laayouni
Department of Mathematics and Statistics
McGill University
January 4, 2007
Dr. Lahcen Laayouni
Advanced Calculus
Multiple Integrals
Double integrals Iteration of double integrals
Multiple
Multiple Integrals
Advanced Calculus
Lecture 4
Dr. Lahcen Laayouni
Department of Mathematics and Statistics
McGill University
January 16, 2007
Dr. Lahcen Laayouni
Advanced Calculus
Multiple Integrals
Spherical coordinates
Triple integrals
Spherical coordi
Math 264 Revised Notes from Class of Sept. 13, 2016
I added on September 14 a few more examples of finding field lines by solving ODEs. Everything else is the same except I removed the part of conservative vector fields which a) we did
not have time to co