Fall 2018

Harvard Extension School
Mathematics E-21a - Fall 2018
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.



HW #6 problems (PDF - due Oct 18 in class or online via Canvas no later than Sat, Oct 20)

HW #7 problems (PDF - due Oct 25 in class or online via Canvas no later than Sat, Oct 27)

Exam #1 took place on Oct 11 during the latter half of class. The exam covered topics through Lecture #5. Solutions will be posted after the exams of all distance students are received. [Note: If you have not already taken the exam, there will be no makeup exam and you will receive no credit for the exam.]

Approximate letter grades for Exam #1
Total points on exam was 54. Median score was 48 (89%).
Mean score was 45.7 (85%). Standard deviation was 8.7.
Scores are available on Canvas.
score grade   score grade
50+ A 35+ C+
47+ A– 32+ C
44+ B+ 29+ C–
41+ B 27+ D
38+ B– 0-26 E
Practice Exam #1     Solutions
Exam #1 solutions will be posted when the
last distance exam is received and graded.

Distance students and other students choosing to take their exam using the distance option MUST make arrangements through the Distance Exams Office at the Harvard Extension School. This means finding a qualified proctor approved by the Distance Exam Office. Those taking a proctored exam using this option will have a 24 hour window during which the exam must be taken. [additional information here]

HW #5 problems (PDF - due Oct 11 in class or online via Canvas no later than Sat, Oct 13)

HW #4 problems (PDF - due Oct 4 in class or online via Canvas no later than Sat, Oct 6)

Regarding Homework: Your worked solutions should be submitted either in class or online via the Canvas site (primarily for distance students but available to all). 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 HW assignments will not be accepted - scans only. Based on class size and practical limits, we may select a subset of each assignment for grading. Solutions to all problems will be posted after the deadline.

Note: Starting with HW #2, there will be a zero tolerance policy for late HW (unless justified by personal or medical reasons). If you cannot submit HW in class you should scan and submit it no later than Saturday at midnight (Eastern time). You are advised not to wait until midnight just in case you have scanning troubles or other computer troubles. Staying current with the assigned HW is strongly advised.

Lectures will meet weekly from 8:00pm to 10:00pm in Harvard Hall 201 and online for distance students beginning Thurs, Sept 6. The recorded lectures should be available within 24 hours of the lecture.

Math E-21a syllabus - Fall 2018     Printable Math E-21a syllabus (PDF) - Fall 2018

TA Renee Chipman will conduct an optional online Q&A session (web conference) on Mondays, 2:00pm-3:00pm (this was changed). There will be time for additional questions from 3:00-3:30pm if needed. Recording will be available.

TA Jeremy Marcq will conduct an optional weekly Q&A session at 53 Church St. #203 on Mondays, 8:00-9:00pm. You may attend this in person or participate online. Recording will be available.

In addition, Robert Winters will hold an informal pre-class Q&A session in Harvard Hall 103 starting ~7:00pm. 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.

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 Note: These Lecture Notes may be edited as the course proceeds

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

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.

The Harvard Extension School is committed to providing an accessible academic community. The Accessibility Office offers a variety of accommodations and services to students with documented disabilities.

Please visit https://www.extension.harvard.edu/resources-policies/resources/disability-services-accessibility for more information.

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 47 48 -- -- -- -- -- -- -- -- -- -- --
Mean score 45.3 43.0 -- -- -- -- -- -- -- -- -- -- --

Important Dates - Harvard University Extension School - Fall 2018

Registration opens for degree candidates Thurs, July 19
Registration opens for all students Mon, July 23
Registration ends Mon, Sept 3
Classes begin (Math E-21a) Thurs, Sept 6
Late registration (with $50 late fee) Tues, Sept 4 to Mon, Sept 10
Course and credit status changes deadline Mon, Sept 10
Course drop deadline for full-tuition refund Mon, Sept 10
Course drop deadline for half-tuition refund   Mon, Sept 17
Withdrawal for WD/WN grade deadline (no refund)     Fri, Nov 23
Final Exam (Math E-21a) Thurs, Dec 20
Grades available online Tues, Jan 8, 2019

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: Friday, October 19, 2018 11:02 PM