Harvard Extension School
Mathematics E-21a - Fall 2024
Multivariable Calculus

• Syllabus
• Mathematica
• Extension School
• Harvard University
• Harvard Math Dept
• Math E-21a staff

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.

OR

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.

Announcements:

Final Exams were received from everyone who were expected to take the Final Exam.
Course grades will be available from the Harvard Extension School on Tuesday, January 7th.
All exams have now been compressed to a manageable size with appropriate resolution.
There are 129 exams to grade in Math E-21a (plus the submitted HW12 extra problem), and 33 exams to grade in Math E-21c.
This will take some time, but I will try to get everything done as soon as possible.
Happy holidays and thank you to everyone who took the course! - Robert Winters

The Final exams are all graded and scores will be available on Canvas.

Math E-21a syllabus - Fall 2024     Printable Math E-21a syllabus (PDF) - Fall 2024     [updated July 22, 2024]

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’ 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.

Course Meeting Times:
Lectures (Robert Winters): Thursdays, 8:00pm-10:00pm [in person (Harvard 201) or via Zoom] or on-demand in Canvas.

All lectures will be recorded and available via the Canvas site for the course. Additional Recitations/Office Hours may be scheduled based on demand.
Optional Recitations: (these may change)

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.

Important Dates - Harvard University Extension School - Fall 2024

Official Academic Calendar

Questions, questions, questions..... and some answers

1) Assignments are initially posted on the working course website: http://math.rwinters.com/E21a/
Announcements and homework assignments will be posted on the Canvas Site for the course, but it's best to first check on this (non-Canvas) site.

2) All of the lectures will be live-streamed and recorded and made available within one day but generally sooner. All of the lectures will be available via a link on the course's Canvas Site for the duration of the course and for a few weeks thereafter.

3) The main materials of the course are the lectures, the weekly Lecture Notes, the Stewart text (best to get an inexpensive one online - links for best deals on the course website), and, of course, the homework assignments.

4) All students will be required to submit HW as a single scanned PDF (easily readable and with a reasonable file size, i.e. not scanned at an unnecessarily high resolution). Classes are on Thursday and the recorded lectures are available no later than Friday. HW assignments are usually due no later than the following Saturday night (approx. 9 days later).

5) We will have two midterm exams and a final exam. The Calendar is given at: http://math.rwinters.com/E21a/calendar.htm. Additional details will follow at a later date. Students can find more information at the Extension School website here: https://www.extension.harvard.edu/exams-online-courses

6) There is an optional Math Placement Test to see for yourself if you're ready for the course. It is not required for registration. There is also a Math Question Center online and in Sever Hall, available to students in a range of math classes.

7) 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).

Regarding Homework: Your worked solutions should be submitted online via the Canvas site as scanned PDF files. A basic standard is that you don't need to scan at resolution greater than about 120dpi. Anything beyond that is not necessary for handwritten work and significantly 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. We will 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.

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).

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.

You are responsible for understanding Harvard Extension School policies on academic integrity(https://www.extension.harvard.edu/resources-policies/student-conduct/academic-integrity) and how to use sources responsibly. Not knowing the rules, misunderstanding the rules, running out of time, submitting the wrong draft, or being overwhelmed with multiple demands are not acceptable excuses. There are no excuses for failure to uphold academic integrity. To support your learning about academic citation rules, please visit the Harvard Extension School Tips to Avoid Plagiarism (https://www.extension.harvard.edu/resources-policies/resources/tips-avoid-plagiarism), where you'll find links to the Harvard Guide to Using Sources and two free online 15-minute tutorials to test your knowledge of academic citation policy. The tutorials are anonymous open-learning tools.

In particular: “To avoid any suggestions of improper behavior during an exam, students should not communicate with other students during the exam. Neither should they refer to any books, papers, or use electronic devices during the exam without the permission of the instructor or proctor.”

“Breaches of academic integrity are subject to review by the Administrative Board and may be grounds for disciplinary action, up to and including requirement to withdraw from the Extension School and suspension of registration privileges.”

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: Thursday, January 2, 2025 11:05 PM