Fall 2017

Harvard Extension School
Mathematics E-21a - Fall 2017
Multivariable Calculus

Robert Winters

Teaching Assistants:
Renée Chipman
Jeremy Marcq

Weekly topics, HW assignments, and supplements
(requires username/password)
Canvas site

The recommended text for this course is Stewart's "Multivariable Calculus: Concepts and Contexts". Either the 4th Edition or the 3rd Edition is acceptable (or an even earlier edition). PLEASE buy a used copy of either edition using the links below.


4th Edition:
ISBN-10: 0-495-56054-5
ISBN-13: 978-0-495-56054-8
(published 2010, Brooks-Cole)

Click on the image
to compare prices.


3rd Edition:
ISBN-10: 0-534-41004-9
ISBN-13: 978-0-534-41004-9
(published 2005, Brooks-Cole)

Click on the image
to compare prices.

Homework assignments are now keyed to the 4th Edition, but all assignments will be made available as PDFs.

A key matching HW exercises in different editions is available on request.



TA Renee Chipman will conduct an online Q&A session (web conference) every Sunday starting at 12:00pm EDT. Recording will be available.

TA Jeremy Marcq will conduct a Q&A session every Tuesday from 8:45pm to 9:45pm in Maxwell Dworkin Room G125 (recording to be available).

In addition, Robert Winters will hold an informal pre-class Q&A session in our regular lecture hall (Harvard Hall 201) starting ~7:00pm or earlier. The purpose of this session is to resolve questions raised after you have struggled mightily with the homework but were either unsuccessful on some problems or have some doubts. Please don't use this session simply to get answers without having dedicated some time on every homework problem.

The Final Exam took place on Thurs, Dec 14 from 7:40pm to 9:40pm in Emerson Hall 105. It was a 2-hour exam. Distance students made the necessary arrangements through the Exams Office.

Approximate letter grades for Exam #1
Total points on exam was 50. Median score was 45 (90%).
Mean score was 43 (86%). Standard deviation was 7.9.
score grade   score grade
46+ A 31+ C+
43+ A– 28+ C
40+ B+ 25+ C–
37+ B 23+ D
34+ B– 0-22 E
Approximate letter grades for Exam #2
Total points on exam was 50. Median score was 47 (94%).
Mean score was 44.1 (88.2%). Standard deviation was 7.8.
score grade   score grade
47+ A 32+ C+
44+ A– 29+ C
41+ B+ 26+ C–
38+ B 24+ D
35+ B– 0-23 E
Approximate letter grades for Final Exam
Total points on exam was 75. Median score was 57.5 (76.7%).
Mean score was 55.0 (73.4%). Standard deviation was 14.5.
score grade   score grade
69+ A 44+ C+
63+ A– 40+ C
58+ B+ 37+ C–
53+ B 33+ D
48+ B– 0-32 E

Scores are posted on the Canvas site (or will be when all distance exams are accounted for).
All Final Exams were scrutinized with great care. Scores are final and not negotiable.

Lecture #1 Notes Lecture #2 Notes Lecture #3 Notes Lecture #4 Notes
Lecture #5 Notes Lecture #6 Notes Lecture #7 Notes Lecture #8 Notes
Lecture #9 Notes Lecture #10 Notes Lecture #11 Notes Lecture #12 Notes
Lecture #13 Notes Lecture #14 Notes -- --

Regarding Homework: Your worked solutions should be submitted either in class or online via the Canvas site (for distance option students). We prefer that assignments be submitted on paper for those who attend class in person. Other students may submit their assignments as scanned PDF files. We will establish limits on file size, but a basic standard is that you should not scan at resolution greater than about 120dpi. Anything beyond that is not necessary for handwritten work and dramatically increases file size. Black & White scans are preferred unless there is a good reason to scan using color. Please also make sure that the contrast is adjusted properly so that all work is legible. Photographs of assignments will not be accepted - scans only. Based on class size and practical limits, we will select a subset of each assignment for grading. Solutions to all problems will be posted after the deadline.

Math E-21a syllabus - Fall 2017     Printable Math E-21a syllabus (PDF) - Fall 2017

Classes meet Thursdays from 7:40pm to 9:40pm in Harvard Hall 201 starting Thursday, August 31.

Teaching Assistant Renée Chipman will also be available for small group sessions and individual assistance on request (if agreeable times can be arranged).

We are offering this course this semester for the first time with an online option. Some details are being worked out as we go, so check back frequently on this site and on the Canvas site for the course.

Note: The "Graduate" credit option is primarily for students enrolled in certain Extension School graduate programs such as the "Math for Teaching" program. All other students (including high school students) should register for the "Undergraduate" option or the Noncredit option (if you will not be submitting homework or taking exams).

Calendar [Topics and homework assignments are posted here.]

Note: At a minimum, you should have successfully completed a full year of Calculus prior to taking Multivariable Calculus. Some topics from first-year Calculus (sequences and series, differential equations) are not essential for this Multivariable Calculus course, but you should be comfortable with all facets of differentiation and integration and their applications.

This course covers the following topics: Calculus of functions of several variables; vectors and vector-valued functions; parameterized curves and surfaces; vector fields; partial derivatives and gradients; optimization; method of Lagrange multipliers; integration over regions in R2 and R3; integration over curves and surfaces; Green's theorem, Stokes's theorem, Divergence theorem.

Please take note that the pace of this course is quite fast and you will be required to do a substantial amount of homework. You should plan accordingly.

Textbook: Multivariable Calculus: Concepts and Contexts by James Stewart - either the 4th Edition (published 2010 by Brooks-Cole, ISBN: 0495560545) OR the 3rd Edition (published 2005 by Brooks-Cole, ISBN: 0534410049). The text should be available at The Coop or via the Internet. Note: If you can find an inexpensive copy of the 2nd Edition of the Stewart text or a comparable text, this should also work fine. Each homework assignment will be posted as a PDF for those with other editions of the text.

Click on the image of your choice of textbook (to the left) for the best online deals.

Accommodations for students with disabilities must be approved and arranged by Rory Stein, Coordinator for Disability Services. Mr. Stein can be reached at 617-495-0977.

Note: The mailbox for Math E21a is located on the second floor of the Science Center, in the corridor off the main outer hallway outside of the elevators heading back toward the plaza area. It's the very first one you'll see as you encounter the mailboxes - at the bottom. However, homework should generally be submitted in class. Any other arrangements and all rules regarding lateness are at the discretion of the Teaching Assistants.

HW #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13
Total points 50 50 50 50 50 50 50 50 50 50 50 50 50
Median score 48 48 46 48 48 48 48 45 47 47 48 48 48
Mean score 44.1 45 41.3 43.5 43.6 44.6 43.2 38.8 39.2 41.8 43 43.6 42.5

Notes on dot products and cross products

Supplement on parameterized curves

Supplement on the Chain Rule and Implicit Differentiation

Supplement on Conservative Vector Fields and the Fundamental Theorem of Line Integrals

Mega-List of Math E-21a techniques       Math E-21a Useful Facts

Here's something: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html

Mathematica (optional, but worth investigating)
If you would like to experiment with Mathematica to graph parameterized curves and functions f(x, y), find level curves, and much more, here are a few labs that will introduce you to this software and show you how to do most of the basic things that are relevant to Math E-21a. You can also cut and paste images produced by Mathematica into other documents and print some pretty dazzling graphics.
Hands-On Start to Mathematica (Flash presentation from Wolfram.com)

Mathematica Lab #1     Mathematica Lab #2     Mathematica Lab #3
[Right-click to save these files, then open them with Mathematica.]

Download your free Adobe Acrobat Reader for reading and printing PDF formatted documents.

Please send comments to Robert Winters.
URL: http://math.rwinters.com/E21a
Last modified: Wednesday, December 27, 2017 12:02 PM