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
Vector and Scalar fields
Conservative vector fields
SUMMARY
Alexandra Tcheng
May 8, 2014
Introduction to vector fields
Def: A vector field associates a vector to each point in its domain.
Important examples:
I
Gravitational vector field: We derived what t
8
3.2. Three Standard Calculation Questions Involving Stokes Theorem. We do example 1 and 2 from the attached pages of Adams and will do (after the break)
Example: Verify Stokes Theorem for
F = h2z, x, y 2 i,
with S is the surface of the paraboloid z = 4
ER 15 Vector Fields
Evaluating Line Integrals = . _ _. . 5,.
The length of (9 Was evaluated by expressing the arc length element :1: =-;. idrI/dtf a;
in terms of a parametrization r ———- 1'03), (.2 5 t _<_: b) of the curVe, and integrating this
fmmtnato
McGill University
Midterm examination
Feb. 23, 2016
Advanced Calculus for Engineers
Math 264
Feb. 23, 2016
Time: 8:35 - 9:55
Prof. R. Choksi
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS
1. If you are not registered in this section, y
Math 264 Notes from Class of Feb. 2, 2016
Summary of Lecture:
Recap of Line Integrals
Recap of the Relationship Between Conservative Vector Fields and Line Integrals
Surface integrals of a scalar function.
1. Summary of Line Integrals (last few classes
Math 264 Notes from Class of Feb. 9, 2016
Flux is very important, so best not be like:
Summary of Lecture:
Reflection on the Sphere
Finish Material from last class notes
Some Examples of Flux Integrals
Announcements:
Midterm is in class on Feb 23. It
Math 264 Notes from Class of Feb. 18, 2016
Summary of Lecture:
Discuss Two Questions From the Last Webwork
Green Theorem Continued
The Divergence Theorem
Correction from the Notes of Last Class: The midterm is 80 minutes NOT 85
minutes.
A correction fr
Math 264 Notes from Class of Feb. 25, 2016
Summary of Lecture:
A few midterm problems
Finish cone problem from the notes of the last class
The Divergence Theorem and The Gravitational Vector Field (from the notes of the
last class)
A Few Optional Comm
Math 264 Notes from Class of Feb. 11, 2016
Summary of Lecture:
A Few Comments on Flux Integrals
The Divergence and Curl of a Vector Field: Definition
The Divergence and Curl of a Vector Field: Physical Interpretation and Examples
Recall the last exampl
Math 264 Notes from Class of Feb. 16, 2016
Summary of Lecture:
Finish Interpretation of Curl Notes From Last Class
Vector Identities
Solenoidal and Irrotational Vector Fields
Start Greens Theorem
The Midterm has 6 questions (a few have several parts):
On
Math 264 Notes from Class of Feb. 4, 2016
Summary of Lecture:
From last class: surface integrals of a scalar function for a parametric surface (notes
from last class).
An example: the torus
Summary of surface integrals of a scalar function
Start surfa
Vector ﬁelds
{Fl c059 d5
F o 'i‘ds
(see Figure 15.8); since 'i‘ = dr/ds is the unit tangent to 6'.
dW = F(r) o'i‘ds = F—dr.
Thus, the total work done by F in moving the object along (3 is
W=fFon32fFodr=f F1dx+F2dy+F3dz.
e (9 e
In general, if F = Fli + sz
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
Greens theorem and its link with the Divergence theorem and Stokes theorem.
1
Greens theorem and Stokes Theorem (easier)
Exercise
Apply Stokes theorem to the following:
The smooth 3D vector field
F~ (x, y, z) = hF1 (x, y), F2 (x, y), 0i
The surface S th
Fluid dynamics
Alexandra Tcheng
This is covered in more details in 16.6 in the textbook.
Contents
Introduction
Vorticity
Conservation of mass
Conservation of linear momentum (not covered)
The Euler equations (not covered)
Introduction
Consider a fluid in
Summary of what you need to get out of the last two lectures
Fourier series & Eigenvalue problems
From the lectures
I expect you to know and understand
the relations between the 3 classical Fourier series,
how they simplify for even and odd functions,
Line integrals of vector fields
Alexandra Tcheng
This is based on Section 15.4 of Adams & Essex, (eighth
edition)
Contents I
Three line integrals
Integral (2)
From Integral (2) to Integral (1)
From Integral (2) to Integral (3)
Independence of parameteriza
Review Part I
Alexandra Tcheng
Contents
From Ch.12 Partial Differentation in Adams & Essex (8th edition)
Scalar functions of several variables (12.1)
Partial derivatives (12.3)
The chain rule (12.5)
Gradients and directional derivatives (12.7)
The differe
Vector and Scalar fields
Alexandra Tcheng
This is based on Section 15.1 of Adams & Essex, (eighth
edition)
Contents
Introduction to vector fields
Definition
Little examples
Why bother?
Illustrations
2D examples
Gravitational field
Electrostatic field
Rota
Math 264 Advanced Calculus for Engineers
Syllabus For Fall 2016
1. General Information
(1) Professor: Rustum Choksi
Office: Burn 1110
Office Hours: Tuesdays and Thursdays 1:00 - 2:30 PM or by appointment
Email: rchoksi@math.mcgill.ca
(2) Prerequisites: Ma
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
Math 264 Notes from Class of Jan. 21, 2016
Summary of Lecture:
Midterm date: Tuesday before or after the break: Feb 23 or March 8?
Introduce gradient vector fields which are also called conservative vector fields
Give a necessary condition for a vector
Slides for the First Math 264 Class on Jan. 12, 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 wher
Math 412 Partial Differential Equations
Exam 1
February 24, 2012
This exam has four questions, each worth 25 points, for a total of 100 points.
1. Use separation of variables to find the solution to the heat equation ut = uxx for 0 x and
t > 0, with bound
The point of this essay is that although Sedaris is portrayed as a snobby child,
he knows when to take action and be there for his family. More specifically, he takes
great responsibility for when his mother is pregnant again. Its almost as if he was
bein
Santander-Torres 1
Vannessa Santander-Torres
Andrew Willmer
345-LPH-MS: Ethics on Stage
February 11th 2016
Position Statement on The Visit
The main issue in this play is that some people value money more than peoples
lives. While very well knowing that ki
Santander-Torres 1
Vannessa Santander-Torres
Sabine Walser
603-LPE-MS: Creative Nonfiction
May 3rd 2016
On my way to Mars
The best days are random: Unplanned and spontaneous. In fact, this is how the
best day of my life began.
July 5th 2013, a hot summer
Santander-Torres 1
Vannessa Santander-Torres
Andrew Willmer
345-LPH-MS: Ethics on Stage
April 18th 2016
Artwork that asks a big question on Vienna by Billy Joel
Artists, throughout many centuries, make art for a reason, and try to share their
ideas or vis