The curve for the final (out of 120 possible points,
15 points for each problem):
We calculated your course percentage using the formula
I looked up the final exam location on class search:
For a practice final you can use
the practice final from last semester.
Problem 3 has a typo.
You need to negate the definition of M.
Otherwise you cannot do the integral in closed form.
So you should have M = -y(1-cos(xy)).
Answers to the practice final:
The final covers chapter 16 and Lagrange Mulipliers.
We are holding the following review sessions
for you all to choose from:
I just posted four pages of
course notes on Gauss's theorem and Stokes'
theorem, which are the culmination of this course.
The two exercises on the last page exercise most of the
essential skills regarding these two theorems
that (based on past semesters) you likely will need
to demonstrate on the final exam.
On Monday at 4:30pm in B115 Van Vleck I am giving a
talk for the Math Club on
what you can do with vector calculus.
I just posted webwork 11. The hard deadline is Wednesday (the day
before the final exam.) It consists of a smattering of problems
from chapter 16.
I just posted webwork 10.
You should complete it by next Friday (December 11).
The hard deadline is Monday night, December 14.
This webwork emphasizes surface integrals.
The first seven problems cover material through
section 16.6.
The last 5 problems involve Stokes' circulation
theorem (section 16.7, which Professor Assadi has
stated and briefly discussed) and Gauss's
divergence theorem (section 16.8).
Stokes' circulation theorem says that the
circulation of a vector field
around the boundary of a surface
equals the flux of the curl of the vector
field through the surface.
(The orientation of the flux and the orientation
of the circulation must satisfy the right hand
rule.)
Stokes' circulation theorem is a straightforward
generalization of the circulation theorem in
two dimensions.
Gauss's divergence theorem
simply says that the outward flux of a
vector field out of a region of space
equals the integral of the divergence
of the vector field
over the interior of the region.
(Recall that the divergence theorem
in two dimensions says that the
outward flux of a vector field out
of a region of the plane equals
the the integral of the divergence
of the vector field over the interior
of the region, so this is a straightforward
generalization to three dimensions.)
In summary, if you really understand Green's
theorem in the plane you will have little difficulty
with its three dimensional generalizations.
I just posted two pages of
notes on surface integrals (§§16.5–6).
I just posted webwork 9.
The hard deadline is Monday night.
You should try to complete it by Friday.
It is based on Green's theorem, which we will cover on Tuesday.
I just posted four pages of
notes on Green's theorem on loop integrals in the plane (§16.4).
Green's theorem is the fundamental theorem of calculus for
vector fields in the plane, and it is
the most critical theorem in chapter 16.
There are two ways of viewing Green's theorem—as
a statement about the divergence and as a statement about
the curl, and the remainder of this course will be devoted
to generalizing this theorem to integrals of vector fields
in three-dimensional space. (First we will have to define
integrals over parametrized surfaces in three-dimensional space.)
If we view Green's theorem
as a statement about the curl, its generalization in three-dimensional
space is called Stokes' circulation theorem, and if we view it
as a statement about the divergence, its generalization in three-dimensional
space is called Gauss's divergence theorem.
As usual, you can make sure that you that you understand these
notes (whether you choose to read them thoroughly or whether you prefer
to rely on the text) by doing the very easy exercises I include.
I just posted six pages of
notes on work integrals and potentials.
These notes explain line integrals, work integrals, and potential
functions and introduce the three differential operators
of vector calculus (grad, div, and curl). They include a few
simple examples and exercises.
I just posted webwork 8.
It consists of 10 problems for sections 16.1 through 16.3.
The hard deadline is the day before Thanksgiving, but you
should aim to finish it by discussion of this coming week.
For chapter 16 I suggest the following exercises from the book
in addition to the webwork:
I just posted
a summary of multivariable integral calculus (Chapter 15). Of course it probably would have been better if I had
written this earlier, but better late than never!
You might find it helpful to read section 3.4,
where I work out an example of transforming the domain
of integration using general coordinates.
I calculated the following answers to the problems in the
practice second midterm. (It's quite possible that I
made a mistake somewhere. If you are confident that your
answer is correct and it differs from mine, please let me know.)
Here are some hints and remarks:
A generic way to find coordinates that make the
equations of the boundary simpler is to regard the
equation of each boundary as the level set of some
function. For example, in section 15.7 problem 22
(which I might have assigned instead of this problem
if I had noticed that this problem is "backwards")
two of the boundary surfaces are determined by the
inequality 0≤xy≤2, so they define a new variable
v(x,y,z)=xy.
On Thursday I will give an in-class review session for
the second midterm.
You can expect midterm 2 to cover chapter 15 and two topics
from chapter 14: multivariable Taylor expansions and Lagrange
multipliers for two constraints.
To prepare, please do the following:
In preparation for the midterm on Tuesday, I suggest that you
finish the problems I assigned for chapter 15, finish the
webwork through webwork 7, work the example problems in
the book as exercises, and write up a summary of the
essentials from chapter 15.
You will be allowed to bring
a single 6"x4" notecard to the exam with writing on
both sides.
I will entertain questions on any of this material on Thursday.
Scores on the first midterm were generally high.
Two people got the maximum score of 140.
To give you an indication of where you stood
with respect to the standards of the course
and the rest of the class, here is a curve:
I just assigned webwork 7, which you should complete by Friday,
November 13, although the hard deadline is Monday November 16
(the night before the second midterm).
Problems 11 and 12 require you to find the centroid of a region.
The centroid is just the center of mass assuming that the
density function has a uniform value (e.g. 1). The centroid of
a region is defined on page 1110 of the text to be the point
whose x-coordinate is the (possibly weighted) average of the
x-coordinate over the region.
I just posted webwork 6, which you should complete by Friday
November 6th, although the hard deadline is not until Wednesday,
November 11th. It covers sections 14.8 through 15.5. We plan to
cover areas, averages, and centers of mass (sections 15.2 and
15.5) on Thursday, so after Thursday's lecture you should have
all the tools to tackle these problems.
We will return the exams on Thursday at the beginning of class
and go over the problems. You should look over your exam and
raise any issues when you receive your exam. Once you leave with
your exam your scores are set.
If you wish to appeal your grade on a particular problem,
you should write an appeal and submit it with your exam
to a person who graded a problem you are appealing.
When you submit your exam to me for reconsideration you are
asking for reassessment. I generally don't change a score,
unless I judge that I made a mistake, in which case I will
increase or decrease your score appropriately.
I just posted webwork 5. It consists of 15 problems for section 15.1.
For chapter 15 I suggest the following exercises from the book
in addition to the webwork:
On Tuesday I will give an in-class review session for the first midterm.
To prepare, please do the following:
I will entertain questions on Tuesday.
I will not work the problems unless you ask me to.
To prepare for the midterm next week, I suggest that you
do the following:
Yesterday I posted webwork 4. The problems use concepts
in sections 14.7 through 14.10. We have already covered
the material for problems 1-5 and problem 10. Next week we
will cover the material in section 14.8,
constrained optimization (i.e. Lagrange multiplers).
The remainder of this note is a brief explanation of this
idea for the case of a single constraint/multiplier.
Constrained optimization and Lagrange multipliers
I just posted webwork 3. It covers sections 14.4-14.7. So you
should be able to get started right away. We will be covering
section 14.7 this week. The hard deadline for for webwork 3 is
Thursday, October 15, but you should aim to finish it before
October 12th.
You should be finishing up webwork 2 (even though the hard
deadline is not until next Friday, October 9).
I just posted
four pages of course notes with exercises for the first half of Chapter 14.
These notes are for the material that we began to cover yesterday
and will cover in the next two lectures (Tuesday and Thursday).
These are the most fundamental concepts in the course.
Please print out these notes, read through the material, and
work the (trivial) exercises.
We have already covered the material in sections 1-4 of these notes.
We will cover the remaining sections this coming week.
Professor Angenent has written up some notes for differentiation
of multivariable functions, which contain essentially the same
content as sections 14.1--14.5 in the Thomas text.
I have posted them on the
course notes page.
I found them to be exceptionally clear in introducing basic
concepts of functions of multiple variables.
Before each lecture you should read over the text and
identify the important definitions and concepts.
After each lecture you should:
I suggest the following exercises from Thomas chapter 14:
I just posted webwork2. It begins with 7 problems of review, so
you should be able to get started right away. You should be able
to do all the other problems after Tuesday's lecture.
Note that the number of attempts on the multiple choice
problems is limited.
You should be able to do the webwork exercises without great
difficulty. If you are only doing the webwork exercises, you
are not doing enough exercises and you will probably do poorly
on the exam. You should be doing the other exercises as well,
including the exercises in the posted course notes and the
exercises in the text that I have mentioned.
For webwork 1, I changed the due date to Friday. (The date in
Moodle is the one you should go by.) But you should finish
webwork 1 by lecture tomorrow. You should not wait until Friday
to finish. I don't plan to respond to questions on webwork on
Friday. If you are still stuck on webwork 1, you might want to
check the hints again; I updated them.
I have posted a syllabus for the course.
Note the dates of the first midterm (Thursday Oct 22)
and the second midterm (Tuesday November 17). Both midterms
will be held in class.
Check out
the wikipedia page on
Differential geometry of curves.
At the bottom is the idea of the Frenet frame that Professor Assadi
mentioned.
In three dimensions, the Frenet frame is the ordered triple
T, N, B, where
T is the unit tangent,
N is the unit normal vector, and
B is the unit binormal vector.
It is easiest to specify the properties of a curve
using an arc-length parametrization. Consider an
arc-length parametrization r(s).
Denote derivatives with primes.
Then
T is r',
N is the unit vector in the direction T', and
B is T × N.
The curvature is defined to be
κ=T'•N.
The torsion is defined to be
τ=N'•B.
Since
T•N=0,
T'•N+T•N'=0, i.e.,
T'•N = -T•N' = κ.
Similarly, since
N•B=0,
N'•B = -N•B' = τ.
I just posted
four pages of course notes on Chapter 13 (curves) with exercises.
We have already covered the material through section 5, so
please read through section 5 and do the exercise at the end
of that section.
After tomorrow's lecture finish reading these notes and
do the exercises at the end.
We are done reviewing chapter 12.
For the next three lectures we will cover chapter 13
(curves in space):
After lecture you should:
I suggest the following problems for chapter 13:
Your discussion grade will be based primarily on
WeBWork. WeBWork is an on-line homework system used
for math courses.
You will access webwork through a generic course-management system called
moodle.
Moodle will display embedded webwork windows,
with their own separate scroll bar.
To log into moodle, point your browser to
http://math.wisc.edu/moodle.
In the very upper right hand corner you should see a tiny link that says
(Login).
You will log in with your NetID and Password, just as for
http://my.wisc.edu.
You should see a large blue link titled
MATH 234 Calculus--Functions of Several Variables, Fall09 (Lec 4).
When you select on this link you should see your moodle page for
this course. In the center of this page you should see a heading
label, "Weekly outline". Under this you should see the
first assignment, titled
webwork1. Click on that link.
You should then an embedded scrolling list of 15 problems.
(Actually, you can click on any of the hyperlinks in this paragraph
and after entering your login information you should be taken
directly to the relevant page.)
When you are in moodle, you should always see a
"breadcrumbs" trail near the top of each page
window which says something like
You should notice a link to hints for webwork1 next to the webwork1 link.
Note that you you must click on "Submit Answers"
to get credit for your answer. You can get partial credit
on problems that have multiple data fields. You can attempt
each problem multiple times.
I urge you to get started.
The first ten problems are review from chapter 12.
We will assign roughly one webwork
assignment each week.
Most webwork assignments
will be due Friday night at 11:59pm. After that time
you will no longer be able to work on the problems for
that assignment. You should not wait until the due
date to complete the assignment! We will let you
know a target date when you should complete the assignment.
The hard deadline is set back from the target due date
to give extra time to people
who experience difficulty with certain problems.
Webwork2 will be related to chapter 14.
I have not posted it yet.
I just posted
four pages of course notes that cover prerequisite
vector and linear algebra. To prepare for Tuesday's lecture,
please print out these notes, read them, and mark anything that
you don't understand or want more clearly explained. As a check
on your understanding, do the exercises under section 2.5.
Topics that I did not discuss in the notes are the determinant,
Taylor Series, and coordinate systems. We will need Taylor Series in
chapter 14, and coordinate systems and the determinant in chapter
15.
Critical prerequisites for math 234:
We will be using Thomas' Calculus, 11th edition.
The course covers chapters 13 through 16:
The topics covered per review sessions are found in the text as follows:
grade cutoff %ile %receiving
A 108 81 19
AB 105 76 5
B 95 56 20
BC 92 51 4
C 78 31 20
D 65 15 16
F 15
Answers to the final exam are here.
final_score = 40% * final exam percentage
+ 20% * midterm 1 percentage
+ 20% * midterm 2 percentage
+ 20% * webwork percentage.
The curve for the course percentage was
grade cutoff %ile %receiving
A 89 79 21
AB 86 73 6
B 77 51 22
BC 75 48 3
C 65 25 23
D 54 12 13
F 12
Thu Dec 17 12:25pm until 2:25pm in *** Bascom 165 ***.
You may bring *ONE* (not two) 4" by 6" notecard to the exam
(written on both sides). Graphing calculators are allowed.
Laptops are not.
Problem 1: greatest value: 2/(1-2*sqrt(2))^2
smallest value: 2/(1+2*sqrt(2))^2
Problem 2:
a. (ye^x+2z, e^x-sin(y), 2x)
b. (4xz,e^z-4yz,0)
c. ye^x+cos(y)
Problem 3: 6*pi
Problem 4: 2*pi*a*(h_2 - h_1)
Problem 5: 16*pi
Problem 6: 2*pi/3
Tue 6:00-7:00pm B130 Van Vleck (Matija)
Wed 7:00-8:30pm B130 Van Vleck (Alec)
grade cutoff %receiving
A 81 25
AB 79 6
B 66 25
BC 65 3
C 54 24
D 42 9
F 7
topic section problems
=================================== ======= ===============================
line integrals 16.1 1-8, 13, 15, 23, 25
work, circulation, and flux 16.2 11, 12, 13, 21, 23(a), 29(a,b), 33
potentials and path independence 16.3 1,3,5,7,9,13,17,19,25,34,37
Green's theorem in the plane 16.4 3,5,7,15,17,22,29,35,39,40
surface integrals 16.5 1, 3, 5, 21
parametrized surfaces 16.6 1, 9, 35
Stokes' theorem 16.7 1, 5, 7/9, 11, 12, 15, 19
divergence theorem 16.8 5, 7, 12, 17, 31, 32
Problem 1: 26/15
Problem 2: 5 + y2 + xy2 + y3 + x2y2/2 + xy3 + y4/2 + H.O.T. (higher order terms)
Problem 3: minima occur at (2,0,0) and (0,2,0) with value f=4;
maximum occurs at (-√2,-√2,2(1+√2)) with value 16+8√2
Problem 4: (3/2)ln(2)
Problem 5: mass=20, center of mass = (22/15, 26/15)
Problem 6: 22/3
Problem 7: πa3(2+√3)/3
Problem 8:
part 1: 76πc
part 2: the answer in part 1 minus the mass of the
slice of the sphere at the top. Using cylindrical
coordinates I got an answer with some horrid fractions.
If you think you might have it right, email me your
answer (reducing fractions to lowest terms)
and I can check it against mine.
(total mass) = ∫∫(mass density)dV
xaverage = ∫∫x(mass density)dV/(total mass)
yaverage = ∫∫y(mass density)dV/(total mass)
cutoff_percent grade cutoff_score percent_receiving
92 A 129 26
90 AB 126 9
82 B 115 26
80 BC 112 3
70 C 98 16
60 D 84 10
F 0 10
topic section problems
=================================== ======= ===============================
double integrals 15.1 1,5,7,11,15-25(odd),31,43,47,59
area and averages 15.2 3, 11, 16
polar integrals 15.3 3, 5, 7, 21, 29, 40
triple integrals 15.4 7, 8, 21, 23, 25, 43
mass 15.5 17
cylindrical and spherical integrals 15.6 1, 11, 13, 31, 39
substitution of variables 15.7 1, 9, 21
sectn topic problems
===== ============================== ====================================
14.1 multivariable functions 3, 9, 13--18
14.2 limits 1,9,13,17,21,27,29,31,33,35,37,51,57
14.3 partial derivatives 7, 13, 19, 33, 34, 41, 43, 51, 57
14.4 chain rule 1, 5, 9, 13, 15, 25, 27, 29, 30
14.5 directional derivatives 2, 5, 9, 15, 17, 21, 27, 29, 36
14.6 tangent planes & differentials 1, 3, 9, 18, 22, 37, 38
14.7 extreme values & saddle points 3, 17, 39, 43, 44, 53
14.8 Lagrange multipliers 1, 5, 7, 9, 17, 18, 26
14.9 implicit functions 1, 3, 7, 11
14.10 Taylor's formula 1, 5, 9
14 Chapter 14 Practice Exercises 35, 37
day book sections topics
========== ============= ================================================
Tue Sep 15 13.1, 13.3 arc length
Thu Sep 17 13.3, 13.4 curvature and the unit normal
Tue Sep 22 13.5 components of acceleration
To prepare for lecture, you should pre-read the section(s) we
will talk about, asking yourself:
section 13.1 problems 11, 17, 23, 29
section 13.3 problems 1, 9, 13, 16
section 13.4 problems 3, 6, 9, 19
section 13.5 problems 11, 13, 15
uw-math → MATH234Fall09_1 → webwork1.
These links show your current context and allow you to
navigate back up.
Note that webwork1 is officially due at 11:59pm
on
Problems from webwork 1
Please complete by
1–5
discussion Mon Sep 14
6–10
discussion Wed Sep 16
11–13
discussion Mon Sep 21
14–15
discussion Wed Sep 22
Wednesday September 23 Friday September 25.
After the due date
you will no longer be able to work on the assignment.
Math 234: 4:30-5:30 p.m. Room 5208 Social Science
* Wednesday September 2 : Vectors, Dot and Cross Products
* Thursday September 3 : Parametric Equations, Lines and Planes
* Tuesday September 8 : Polar, Cylindrical, and Spherical Coordinates
These review workshops cover the algebra of multiple variables.
We will be studying the calculus of multiple variables.
Calculus builds on algebra.
chapter basic topic mathematical object of study
======= ============= ======================================
13 curves functions from R to Rn
14 surfaces derivatives of functions from Rn to R
15 volumes integrals of functions from Rn to R
16 vector fields functions from Rn to Rn
day subject where is this covered? when will we use this?
=== ============================= ====================== ======================
Wed Vectors, Dot and Cross Products 12.1--12.4 chapters 13 and 16
Wed Parametric Equations 3.5 chapter 13 and 16
Thu Lines and Planes in Space 12.5 chapter 14
Tue Polar coordinates 10.5 chapter 15
Tue Cylindrical, Spherical Coordinates 15.6 chapter 15
To see whether you need the first two review sessions,
start on the first assignment:
section and topic page problems
================================= ======== ================================
* 12.5. Lines and Planes in Space (p. 889): Problems 63, 64, 65, 66, 67, 69
* Ch. 12 Practice Exercises (p. 900): 18,22,24,32,36,42,43,49,52,61,62
* Ch. 12 Advanced Exercises (p. 903): 7, 10, 11
[We will not be collecting this assignment.]