MATH 200: Multivariable calculus, the common website.
Winter Term
2 2019/20
- Textbooks:
-
CLP-3 website (contains
old exam solutions) (by Professors J. Feldman, A. Rechnitzer and E. Yeager).
This will be our primary textbook for reading.
The Problem Book on this website will be our main source of problems (referred to as CLP-problems).
- An additional reference: Apex Calculus
Suggested problems from this book will be referred to as AC-probelms.
The next two references will be used
later in the course:
- Secondary Textbook no. 1 ;
- Secondary Textbook no. 2.
Individual section websites:
-
Section 203
(Prof. Julia Gordon).
- Sections 201, 202, 204 -- please use Canvas.
Course syllabus
Course syllabus and outline
that
describes the
learning goals, etc.
Exams and Marking
Course mark will be based on the Webwork (10%),
four in-class tests (10% each), and the
final exam (50%).
Policies:
All exams are closed book, but you can bring 1 formula sheet written on both sides. Calculators will not be permitted.
There will be no make-up term tests, and no late homework accepted. Students with concessions (e.g. for illness or family emergencies) will have the
weight of a term tests transferred to the other tests and the final exam; the weight of a WeBWorK assignment will be transferred to the other
WeBWorK assignments.
You can receive {\bf one} concession during the term, by submitting the
Academic Concession request Form to your
instructor.
Further concessions or missed final exams need to be discussed with the Academic Advisors of your Faculty.
There cannot be any exception to this university-wide policy.
Webwork will generally close at 11:59pm
on the specified date
(please look at the dates carefully).
No extensions are possible -- please do not ask.
If for any reason you have to miss the final exam, it is the university-wide policy that you need to
apply for "standing deferred" status through your faculty. Missed finals are not handled by the instructors or the
Mathematics Department.
UBC takes cheating incidents very seriously. After due investigation, students found guilty of cheating on tests and
examinations are usually given a final grade of 0 in the course and suspended from UBC for one year.
Note that academic misconduct includes misrepresenting a medical excuse or other personal situation for the purposes
of postponing an examination or quiz or otherwise obtaining an academic concession.
Homework
- On Canvas, you will have two different pages for Math 200: "Math 200 ALL" and Math200-your-section-number.
The common page "Math 200 ALL" will have the links to webwork, piazza, and some course-wide announcements.
Your section's page will eventually have your quiz and midterm marks.
All other content will be posted here and on the individual sections' web pages (not on Canvas unless specified by your instructor).
- Homework assignments should be submitted online through Webwork.
To access Webwork, log into Canvas, go to the Math 200 All page (NOT your section's
page), and click on "assignments" on the left. You should see only one item "Webwork link".
Click on that link; it should open Webwork in a separate window. Then work with Webwork as
usual.
FAQ about Webwork
advice on starting webwork early
- Please use Piazza
as the main resource for help with webwork-related and other questions.
It is a forum, which will be monitored by our TA, where you can post questions and answers about webwork.
Please use the "e-mail instructor" button in webwork *only* if the question is not answered on Piazza,
and you posted it and did not receive an answer.
Sign-up link for our class (all sections) on Piazza (you need to individually sign up using this link, we have not
imported your email addresses to Piazza to protect your right to privacy).
Piazza should also show up on your main menu (on the left) on the Canvas page that is COMMON to all sections.
Getting help
- In addition to your instructor's office hours, please take advantage of the
Math Learning Centre drop-in tutoring.
Do not wait till the exams -- if you feel uncomfortable with any of the material, talk to your classmates, talk to the instructor, and come ask questions at the Math Learning Centre.
Resources
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 many problems in Webwork which require thinking and which Wolfram Alpha cannot do; for the more mechanical ones that it can do, if you just use the software and copy the answers, it detracts from your learning. You might get a few extra points for the webwork problem, but you'll certainly lose much more on the exam for not having that skill. So use this great software to your advantage (to help you visualize the objects we study, and to learn), not to your disadvantage (to cheat on Webwork).
Math
Learning Centre drop-in tutoring.
Announcements:
- The course is moving online starting March 16! Please still be
avialble and near a
computer during your sual lecture times. More communications about the
details will be sent by Monday March 16. Please check your email (and
Canvas) for further announcements.
- Test 4 will happen on march 25/26. Details to be announced later.
Weekly learning goals for the online portion of the course
This is a detailed list of things you need to be able to do and understand
by the end of each week. Some resources are listed below. More resources,
lecture recordings, etc. will appear on Canvas as we progress.
Feel free to find and share extra resources on Piazza.
After the first few items, the words "need to" are dropped.
The learning goals are labeled as "skills" and "concepts" to alert you
whether it is something you need to be able to do vs. something you need
to ponder, internalize, and see why it makes sense. Both are important
and will be tested.
- Week 1, March 16-20:
- Double integrals over general regions in the plane:
- (Skills): Need to be able to write a double integral over a domain
as an
iterated integral; conversely, given an iterated integral, need to be
able to sketch the domain of integration that this integral represents.
- (Skills): Need to be able to switch the order of integration in a
double
integral (note that as you do that, it might have to be split into a sum
of two or more double integrals).
- (Concepts): Areas, volumes, mass, centre of mass:
- need to recognize that a double integral over a domain D of a
positive
function f(x,y) computes the volume under the graph of f(x,y) and above
the region D
- Need to *understand* that if the function f(x,y) is constant and
equal to 1 everywhere on D, then its double integral over D simply equals
the area of D.
- understand the concept of the centre of mass of a 2-dimensional plate
("lamina") and know the formula for finding its coordinates.
- Polar coordinates:
- (skills and concept) Need to know the conversion formulas between
Cartesian and polar
coordiantes
- (Skills) Be able to convert a double integral of a fuction f(x,y)
over a
domain into polar coordiantes for appropriate domains.
- (Skills, concepts) Know how to write an equation for certain curves
in polar coordinates
and also (ideally) how to sketch a given equation in polar coordiantes
(this
includes cardioids, "rose" curves, and circles that pass through the
origin.
Should be able to set up integrals over the domains bounded by such curves
(even if cannot sketch the curve).
Resources for this week:
- Lecture notes from an earlier year:
- Lecture 1: double integrals
(for most students this will be review of last week, but might still be
useful).
- Lecture 2:
Interchanging the order of integration in a double integral; starts polar
coordinates. (Again mostly review of the last class of last week for most
sections).
- Lecture 3: Polar coordinates; 6-petal rose
in polar coordinates; using double integral in polar coordinates to
compute area.
- Lecture 4: Mass and centre of mass of a
lamina.
- Reading from CLP-III: 3.1, 3.2, 3.3; Reading from Apex calculus:
13.3, 9.4. See the syllabus (below) for suggested problems.
- Summary of what you need from Math
101.
Week 2, March 23-27:
Week 3, March 30 - April 3:
Week 4, April 6-8:
Review materials for Test 4
Topics covered:
- Lagrange multipliers (only one constraint).
- Double integrals: writing an integral over a domain as an iterated
integral; changing the order of integration
- double integrals in polar coordinates.
- Applications of double integrals: finding area of a domain; average
value of a function of two vraibles; total mass and centre of mass.
-
A summary of integration techniques from math 101.
- Final from 2003, with solutions .
Look *only* at problem 5.
- More coming soon.
Older review materials and announcements
Test 3 review materials:
Topics covered:
- gradients, directional derivatives, tangent planes
for general surfaces of the form F(x,y,z)=0; steepest ascent/descent, the
direction of the fastest increase of a function.
- Critical points, classification of critical points;
- absolute
maximum/minimum of a function on a closed bounded domain (Section 2.9) --
not including Lagrange multipliers.
Review materials for Test 3
(Approximate) week-by-week course outline
Chapter numbers in the description of the material are from
CLP-3 Calculus ; chapter
numbers in the
suggested problems are from
Apex Calculus unless otherwise
specified. Please note that this is only an approximate outline; it may be updated as the course progresses.
Please also check the individual sections' websites for more specific information about your lectures.
Some illustrations and supplemental materials may be posted below the description of a week's lectures, please keep checking.
- January 6-10:
1.1 Three-dimensional coordinate systems; distance between points in space.
1.2.1, 1.2.2: Vectors; basic operations with vectors; length of a vector, equation of a sphere in
space, unit vector in a specified direction.
1.2.2: Dot product;
Using dot product to find an
angle between lines. Optional reading: application to finding forces (1.2.3).
Suggested problems:
AC 10.1: 1-3, 7, 9, 12, 16
AC 10.2:
1-5, 8, 11, 15, 20, 23, 27, 31
AC 10.3: 1-3, 11, 15, 19, 31, 39.
- January 13-17:
1.2.5. Cross product. Using cross product to find a vector orthogonal to two
given ones; cross product and area.
Mandatory reading: 1.2.4 -- area of a parallelogram.
Starting sections 1.3-1.5: Equations of lines and planes in space.
Webwork 1 due on January 13.
Suggested problems:
10.4:
1-5, 9, 15, 27, 30, 31, 35, 39, 41.
- January 20-24:
1.3, 1.4 and 1.5: Continuing with Equations of lines and
planes.
Symmetric and parametric equations of a line in space.
Equations for planes in space.
Equations for a line of intersection of two planes, etc.
Finding distances in space: distance from a point to a plane, etc.
1.7, 1.8. 1.9 : Cylinders and quadric surfaces.
Homework 2 due.
Suggested problems:
AC 10.5: 7, 11, 21, 27, 31.
10.6: 1, 2, 9, 11, 14, 15, 17, 19, 25, 29, 32;
- January 27-January 31:
12.1 in Apex Calculus: Functions of several variables. Domain and range. Level curves and level surfaces.
Brief dicsussion of limits and continuity for functions of two variables.
Section 2.1 in CLP textbook (including 2.1.1).
Homework 3 due.
Test 1 on vector geometry.
Suggested problems:
10.1: 15, 17, 23-26, 27, 32.
12.1: 1-6, 7, 11, 17, 19, 21, 23, 26, 27, 29, 31
- February 3-7.
2.2, Partial derivatives; 2.3 higher-order partical derivatives.
Tangent planes (section 2.5 up to (not including) 2.5.2)
Linear approximations: 2.6 (up to (not including) 2.6.1).
Homework 4 due.
Suggested problems:
Section 12.3: , problems 1-4, 5, 13, 19, 29, 33.
Section 12.4: 7, 10, (find equation of tangent plane to z=f(x, y) at given point for 11, 12) , 13, 15, (find linear approximation for 17, 18 at the given point).
- February 10-14.
2.4 Chain rule and the implicit differentiation examples from 2.2.
The gradient and directional derivatives: start 2.7 (not including
"optional"
examples).
One additional topic to recall here: parametric equation of a segment connecting two points A and B.
Homework 5 due.
Suggested problems:
Section 12.5: 1-5, 9, 17, 21, 29.
- February 24- 28.
2.7 Directional derivatives and gradients, continued.
2.5.2 Gradient as a normal vector to a level surface.
Tangent planes to level surfaces.
Tangent planes to graphs of functions of two variables, revisited.
Test 2 on differentiation, domain/range, level curves
Homework 6 due.
Suggested problems:
Section 12.6, problems 1-6, 13, 15, 21, 23, 25, 27
Section 12.7, problems 17, 19, 21, 23
- March 2-6.
Section 2.9 Critical points: the second derivative test,
absolute maximum and minimum values.
Lagrange multipliers (Section 2.10 (without the optional 2.10.1)).
Suggested problems:
Section 12.8, problems 1-4, 5, 7, 11, 13, 15, 17 (also 11, 13, 15, 19 from 14.7 in secondary text #1)
Section 14.8 (from secondary text #1) 5, 10, 11, 12, 13, 15, 17
- March 9-13:
2.10 Lagrange multipliers, continued. (two constraints not included).
Starting integration: 3.1 (the definitions; area; integral of a function of two variables over a rectangle.
Iterated integrals (over a rectangle).
Fubini theorem (without proof).
Test 3 (differentiation 2)
Homework 7 due.
Suggested problems:
see above for 14.8, see below for 13.1
- March 16-20:
Double integrals over general regions.
Interchanging the order of integration. Sections 3.1, 3.2, 3.3
Polar coordinates (See also Sections 13.3 and 9.4 in Apex Calculus ).
Averaging a function of two variables over a domain; Mass and centre of
mass.
A summary of
integration techniques from Math 101.
Homework 8 due.
Suggested problems:
13.1: 7, 9, 19, 21 (also see #3, 5, 10, 13, 15 from section 15.1 secondary text #1)
13.2: 1-4, 7, 9, 13, 17, 21, 25 (also see #17, 21, 23 from section 15.1 secondary text #1)
13.3: 3, 4, 8, 13;
- March 23-27:
3.5 Triple integrals. Six different ways of writing a triple
integral as an iterated integral. Applications. Triple integrals in
cylindrical coordinates. (Section 3.6 in CLP-III)
Test 4 on double integrals, and
Lagrange multipliers.
Homework 9 due.
Suggested problems:
13.4: 1, 5, 6, 13, 24;
13.6: 5, 7, 9, 11, 13, 15, 19, 23.
- March 30- April 3:
Triple integrals in cyindrical coordinates, continued (Section 3.6)
Triple integrals in spherical coordinates (Section 3.7).
See also
14.4 (from secondary text #2). Note: look at Section 14.4 only up to
the end of Example 7 on p. 544 (after that it is interesting reading, but
we are not covering it in class).
Homework 10 due.
- April 6-8:
i Triple
integrals in spherical
coordinates, continued (Section 3.7).
Review. See also
14.4 (from secondary text #2 --
see the note above as to
where to stop).
Suggested problems:
14.4 (from secondary text #2): 11, 13, 15, 19, 22, 23
Homework 11 is due.