MATH 226 Advanced Calculus III: Multivariable.
Fall semester 2015
Instructor: Julia Gordon.
Where and when : MWF 11am-noon, in BUCH A 202.
My office: Math 217.
Office hours: TBA.
e-mail: gor at math dot ubc dot ca
- Text: Adams, Essex, Calculus (several variables) (edition 8).
The course will be based on Chapters 10, 12, 13, 14.
- Course outline . Please note: this is an honours course.
Homework
Most assignments will consist of some written problems to hand in and
Webwork (Find
Math 226 in a long list of courses and click on it).
You might also be able to access Webwork via
Connect.
For Webwork syntax (i.e. how do type such-and-such function, for
example, sqare root, in Webwork), click
here
.
- Homework 1. Written part due Friday
September 18 in class:
-
Homework 2. Due September 23.
- Webwork
Problem Set 2 is now open (due Wednesday September 23 at
11:59pm).
-
If you are uncomfortable with proofs, set notation, quantifiers, etc., please read this handout and do the questions in it.
If you are uncomfortable with the statements that have several quantifiers (e.g. "for every x exists
y such that..."), please read this as
well.
Do not hand in any of these exercises, but feel free to come and discuss them with me.
-
Homework 3, due September 30:
- Webwork Problem set 4, due October 7.
-
Webwork Problem set 5, due October 15! (Note: unusual due date -- it's a
Thursday
evening).
- Written assignment (extended to Monday October
26).
Note: there was a typo in Question 4; now it is fixed. The limit in this
question should be lim_{x -->0} (the limit as x approaches 0, not the
limit as (x,y) approaches (0,0)). Clear the cache of your browser to see
the new version.
There is also Webwork 6 due October 21!
Solutions
- Take-home midterm. Due November 18.
- Webwork 7 is due Wednesday October 28 (extended to Friday October
30).
explanation of the error in the
Webwork's solution to Problem 18(b), and correct solution.
- NO homework is due on November 4, but there is "PracticeProblems" set
in Webwork
that
I highly recommend to help prepare for the midterm. (You do not have to
do all the problems of course, and you will get no credit for them).
Solutions to this set in Webwork will appear at 1:01pm (in the early
afternoon) on Tuesday Nov.3,
so that you can look at them before the exam.
- Webwork 8 due November 12.
-
No webwork due November 18, but there is home reading: 12.9 - 13.3 by
Monday November 15. Take-home midterm is due November 18.
- Due November 25:
- Due the last week of
class and after:
- The last written assignment: Section 14.3, Problems 11, 21, 12, 29.
Due
Friday December 4 (the last day of class).
- Webwork 10 due December 2.
- Webwork 11 due December 9. -- This Webwork is optional: the score for
it, out of 1 point, will be added to your final mark as a bonus. You need
to do it to be well prepared for the final, though.
Resources
Interactive graphic demos by Joseph Lo.
These are very helpful illustrations to some of the concepts we study in this course. The individual graphics from this site
will be linked from the relevant weeks in the week-by-week course description below.
You can use Wolfram Alpha -- it is a wonderful tool for plotting graphs of functions of two variables, for example. If you want to visualize, for example, the surface x^2+xy-y^2+3z=0, just type in "plot (x^2+xy-y^2+3z=0)".
A note about Webwork and Wolfram Alpha: there will be problems in Webwork
which Wolfram Alpha cannot do; for the more mechanical ones that it can do, if you just use the software and copy the answers, make sure you know how to do such problems by hand. Doing a few "drill" problems is occasionally necessary.
If you feel that Webwork gets too repetitive, I prefer that you e-mail me that feedback rather than keep crunching it
on Wolfram Alpha.
Math
Learning Centre drop-in tutoring.
Announcements:
- Review session: Wednesday December 9, 3-5pm, in Math 203.
- Office hours after the end of term: December 10 and 11, noon-2pm.
- The final exam is Saturday December 12, 8:30am.
One-sided size A4 formula sheet
is allowed on the exam!
Please remember to fill out teaching evaluation forms.
Review materials for the Final Exam
- Detailed list of topics
- Old exams can be found at Department website
(scroll down to Math 226).
- "Integration Practice" set in Webwork.
- Now you can do all the problems in Sample Midterm 1 and Sample
midterm 2, and Final exams from 2012 and 2013 linked below under "midterm
review" for Midterms 1 and 2. I highly recommend doing
that!
- Notes from the review session on
Dec. 9 (the last 3 pages repeat the notes starting from the "Implicit
function" topic on pp. 4-5, but by different note-taker. There was no
document camera, so thanks to the two note-takers!).
Review materials for Midterm 2
Most midterms listed below have solutions; DO the problems yourself,
without reading the solutions. Read the solutions only to check your
answers.
Reading the problems with solutions right away does not help to prepare at
all!
Resources for Midterm 1
- Midterm will cover: Sections 10.1-10.4 and 12.1.
You are allowed to bring a formula sheet (single side, A4 or standard
notebook page).
- Detailed list of topics . (please note:
this was updated on October 3 to remove limits; contour plots are still
in).
-
2010
final (do only Problem 1);
- An
old midterm (Only look at Problems 1(a)(b)(d) and 4 -- everything
else is not on our exam).
Note: in that version of Math 226, they covered material in slightly
different order, and open/closed sets were discussed later, but partial
derivatives discussed earlier (also, this year the term started late);
because of this, only look at the specified problems in the above exam.
For practice problems with open/closed sets (which are on our exam),
please see:
-
Final exam 2013 (Do only Problem 7); Solutions
- Final exam 2012 (Do only problem 2); Solutions
(Approximate) week-by-week course outline
Pointers to the text, topics covered, and other references will be posted here as the course progresses.
- Sept. 9-11:
Sections 10.1. Coordinates in space; open and closed sets and
equations/inequalities defining them.
scan of lecture 2 .
- Sept. 14-18:
Sections 10.2 -10.4. Vectors, dot product, cross product, projections.
Open and closed sets, boundary points (end of 10.1).
- Monday lecture
.
- Wednesday lecture (this lecture was almost
entirely devoted to the discussion of open/closed sets, interior
and boundary points from the end of 10.1); at the end, finished 10.2
(projections).
for Friday: please read 10.3, start reading 10.4
- Friday lecture . We defined cross product and
gave
a formula for it in terms of components of the vectors.
Next class, will finish explaining why this formula works.
Also, defined triple product and talked about its relation with volumes.
Reading for next class: 10.3, 10.4.
- Sept. 21-25:
Sections 10.3-10.4, and 12.1-12.2. Also please read 10.5.
- Monday lecture (equations of lines and
planes).
Reading: 10.3, 10.4, 10.5 (10.5 will not be discussed in class, but please
read it).
- Wednesday lecture (more about equations of
lines; distance from point to plane, point to line, between lines).
Next class: starting 12.1!
- Friday lecture . (Functions of several
variables; level curves). Here are the pictures that were discussed: the
main contourplot of the "mystery function" from the notes.
The same function with different zooming
and
more
contours ; Same plot zoomed in on the
bad point.
- Sept. 28- Oct.2:
Sections 12.2: limits and continuity.
-
Monday lecure
(The epsilon-delta definition of a limit.)
When these notes refer to
contour plots, these are the plots posted above from last week Friday
class.
-
Wednesday lecture . Epsilon-delta, continued: one
example worked out.
Friday lecture .
Continuity -- definition and properties of continuous
functions (properties without proof, for now).
Then started discussing examples where one needs to have a good definition of a limit -- namely, a function whose limit
along both the x-axis and the y-axis at the origin exists, but a limit at the origin as a function of two variables
does not (this is basically the same example as discussed in 12.2).
- Oct.5 - Oct. 9: Midterm; limits and continutity, continued.
- Monday: Midterm 1 . Solution to problem 2 . Solution to Problem 3 . the rest of solutions .
- Wednesday lecture . Epsilon-delta definiton, continued: an example when a limit along any
line exists, but the limit doesn't exist. Also, one more example when the limit at the origin exists.
- Friday lecture . More on epsilon-delta: proof of one of the properties of continuous
functions: if f(x), g(x), and h(x,y) are continuous, then F(x,y)=h(f(x), g(y)) is continuous. This finishes the
discussion of epsilon-delta definitions, and 12.2.
Defined partial derivatives (started 12.3).
- Oct.14 - Oct. 16: Partial derivatives; linear approximations.
Please read 12.3 and 12.4. We will go through these sections fast!
Problem set 5 due in Webwork on Thursday October 15!
We also started 12.5 Chain rule.
- Oct.19 - Oct. 23:
Chain rule; implicit differentiation; differentiability, differentials,
linear approximations.
Textbook references (in this order): 12.5 (chain rule), the beginning of
12.8 (implicit
differentiation), then 12.6 (differentiability, differentials); then
started 12.7 (defined the gradient and directional derivatives on Friday).
Webwork 6 due on October 21, as usual.
- Oct.26 - Oct. 30:
Gradients, directional derivatives; geometric meaning of the
gradient. Implicit function theorem.
Reading: 12.7, 12.8.
- Nov.2 - Nov.4:
- Nov.9 - Nov.13:
Will continue 13.1 (Monday will start with the explanation why the second
derivative test works - this would require part of 12.9); then will talk
about absolute (global) maximum/minimum. Please read 13.1 and 12.9.
- Monday: did the blackboard lecture; it was about 12.9 -- derived the
form of the second-degree term of the Taylor approximation for a function
of two variables (which is given by the Hessian matrix).
- Wednesday: Remembrance day.
- Friday lecture. Proof/explanation of the
second
derivative test for the classification of the critical points.
- Read: 12.9, 13.1, 13.2, 13.3.
Will not do 13.2 in class but you should read it.
Next class: 13.3, Lagrange multipliers.
- Nov.16 - Nov.20:
Lagrange multipliers. Starting integration in several variables.
- Nov.23 - Nov.27:
- Nov 30 - Dec 4:
Triple integrals:
