Also PDF brochure form of the syllabus is still in the process of being updated.

Course name

Multivariable Calculus Math 21a,, Harvard College/GSA: Course ID 119196, Exam Group FAS05_A,Fall 2017/2017, This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning (previously called Core area requirement for Quantitative Reasoning).

Course head

Oliver Knill,, SC 432, Harvard University

Meeting time

After a short intro meeting on Wednesday, August 30, 8:30-9:00 AM, Science Center B, which all students attend, classes are taught in sections on MWF 9,MWF 10,MWF 11, MWF 12, TuTh 10-11:30, TuTh 11:30-1. Our "teaching in section" approach has been praised for the last decade. It is more expensive and requires coordination but it has proven to build closer relationships with the instructor, allowing discussing the material in a Socratic manner, also one by one in office hours and have instructors knowing you in person. There are global reviews and a Mathematica workshop which are taught for the entire course. Classes in individual sections start on on Wednesday, September 7th, for the MWF sections and Thursday, September 8th for the TTh sections. More information about sectioning.

Problem sessions

Course assistants will run additional problems sessions. It is highly recommended to use this time for discussing the material.

Office hours

Office hours of all the teaching staff will be posted. You are welcome to join any of the office hours.


The Math question center (MQC) is a place, where you can hang out to work on your course work. The MQC takes place Sun-Thu 8:30-10:30 PM in SC 309. ( The MQC is in 309! We switched rooms with the course Math 1b during the semester) The MQC will open on Thursday, September 7, 2017 and stay open until December 3, 2017.
Prerequisite A solid single variable calculus background is required. The mathematics department provides advising resources if you are unsure. You can also check with the course head of this course.

The course

Multivariable calculus is a fundamental pillar for many other things:

It extends single variable calculus to higher dimensions. You will see that the structures are much richer than in single variable and that the fundamental theorem of calculus generalizes to higher dimensions.
It provides vocabulary for understanding fundamental processes and phenomena. Examples are planetary motion, economics, waves, heat, finance, epidemiology, quantum mechanics or optimization.
It teaches important background needed in social sciences, life sciences and economics. But it is rigorous enough that it is also suited for students in core sciences like physics, mathematics or computer science.
It builds tools for describing geometrical objects like curves, surfaces, solids and intuition which is needed in other fields like linear algebra or data analysis. Geometry is currently extremely hot: tomography methods in medicine, computer games, google earth, social network analysis all use geometry.
It relates to culture and history. The quest for answering questions like "where do we come from", "what will future bring us", "how can we optimize quantities" all use calculus. They were the motor to develop it. Euler, the inventor graph theory for example knew geometry and calculus well. The history of calculus contains fascinating stories, starting from Archimedes, 2300 years ago up to the modern times, where new branches of multivariable calculus are developed to understand the structure of nature.
It develops problem solving methods. Examples are optimization problems with and without constraints (which is the bread and butter for exconomics), geometric problems, computations with scalar and vector fields, area and volume computations.
It makes you acquainted with a powerful computer algebra system which allows you to see the mathematics from a different perspective. Such systems are more and more needed for visualization, experimentation and to build laboratories for your own research. No programming experience is required however. We will provide templates and get you started with a workshop.
It prepares you for further study in other fields. Not only in mathematics and its applications, but also in seemingly unrelated fields like game theory, probability theory, discrete mathematics, sociology, or number theory, where similar structures and problems appear, even in a discrete setting. Without geometric intuition and paradigms learned in calculus, it is rather hard to work in those fields.
It improves thinking skills, problem solving skills, visualization skills as well as computing skills. You will see the power of logical thinking and deduction and why mathematics is timeless.


The lecture times are MWF 9, MWF 10, MWF 11, MWF 12, TuTh 10-11:30, TuTh 11:30-1. The sections are all coordinated and teach the same material. Learning it in a smaller class helps you to absorb it better and to learn more efficiently. You will section for this course online. The actual lectures start on Wednesday, September 7. Tuesday/Thursday sections start on Thursday, September 8.


Virtually all multivariable calculus books have the same structure. An example is Multivariable Calculus: Concepts and Contexts, 4 book by James Stewart fourth edition. The edition is not important and even an other book works. Stewart Multivariable Calculus Edition 4E has the ISBN number ISBN-13:978-0-495-56054-8. The version which includes single variable is ISBN-10: 0-495-55742-0. Also newer or older editions work. The Cabot library has a copy on reserve.


There are two midterm exams and one final exam.


First and second hourly     30 % total
Homework                    25 %
Mathematica project          5 %
Final                       40 %
Final grade                100 %

Graduate Credit

This course can be taken for graduate credit for almost all graduate schools. We regularly have graduate students taking our multivariable calculus course. If in doubt, check with your school before the semester starts whether you get credit. To fulfill the graduate credit requirements, a minimal 2/3 score must be reached for the final project.

Academic integrity

The usual rules outlined in the student handbook apply. As outlined in the Grades section, submitted work consists of:

Mathematica project

The course traditionally features a Mathematica project, which introduces you to the advanced and industrial strength, computer algebra system. It is an extremely high level programming language also: where objects can be almost anything: a picture, a book, a website, a social network or a movie. The command
A = Import[""];
n=StringCount[A, "Harvard"];  Speak[A];
for example reads in the syllabus of this course, counts the number of words "Harvard" and reads the text to you. Details and the assignment will be posted later. Mathematica for which Harvard has a site license (currently, the latest edition is Mathematica 11). At the end of the semester you submit a short project. The actual lab will be posted later in the semester. This software does not lead to any additional expenses and the total time for doing the lab is of the order of a homework problem if you do the minimal requirement.


FAS calendar.
Su Mo Tu We Th Fr Sa     Event
         30 31  1  2   0 Aug 30 Intro 8:30 Hall B, Sep 5 Labor day
 3  4  5  6  7  8  9   1 Sep 6: first MWF class, Sept 7 TTh class
10 11 12 13 14 15 16   2 
17 18 19 20 21 22 23   3 
24 25 26 27 28 29 30   4 Sep 27, Hall B
 1  2  3  4  5  6  7   5 
 8  9 10 11 12 13 14   6 Oct 9: Columbus day, no classes
15 16 17 18 19 20 21   7 
22 23 24 25 26 27 28   8
29 30 31  1  2  3  4   9 Nov 1, Hall B
 5  6  7  8  9 10 11  10 
12 13 14 15 16 17 18  11 
19 20 21 22 23 24 25  12 Nov 22 -Nov 26 Thanksgiving
26 27 28 29 30  1  2  13 Dec 1, last day of class
 3  4  5  6  7  8  9  14 Dec 2-Dec 8 Reading period
10 11 12 13 14 15 16  15 Dec  9-Dec 19 Exam period

Day to day lecture

 Hour      Topic                  

         1. Vector geometry               

             Labor day
  1          - coordinates and distance          9.1    
  2          - vectors and dot product           9.2-3  

         2. Functions                          

  1          - cross product, lines planes       9.4-9.5    
             - distances
  2          - level surfaces and quadrics       9.6    
  3          - curves, velocity, acceleration   10.1-2  

         3. Curves                              

  1          - arc length and curvature         10.3-4  
  2          - other coordinates                 9.7    
  3          - parametric surfaces              10.5    

         4. Surfaces                           

  1          - review for first hourly
  2          - functions and continuity         11.1-2  
  3          - partial derivatives              11.3    

         5. Partial derivatives               

  1          - partial differential equations   11.3    
  2          - linear approximation             11.4    
  3          - chain rule,implicit different.   11.5    

         6. Gradient                          

             - Columbus day Oct 10, no classes 
  1          - tangent spaces                   11.4 11.6
  2          - directional derivative           11.6    

         7. Extrema                           

  1          - maxima, minima, saddle points    11.7    
  2          - Lagrange multipliers             11.8    
  3          - Global extremal problems         11.8    

         8. Double Integrals                  

  1          - double integrals                 12.2-3
  2          - polar integration                12.4
  3          - surface area                     12.6    

         9. Triple integrals                  

  1          - review for second hourly       
  2          - triple integrals                 12.7    
  3          - spherical integration            12.8    

        10. Line integral theorem              

  1          - vector fields                    13.1
  2          - line integrals                   13.2
  3          - line integral theorem            13.3    

        11. Green and Stokes theorem        

  1          - Greens theorem                   13.4    
  2          - div, curl                        13.5
  3          - Flux                             13.6    

        12. Divergence theorem               

  1          - Stokes theorem                   13.7    
  2          Thanksgiving break  (no class)     Nov 23- Nov 26
  3          Thanksgiving break  (no class)

        13. Integral theorems Overview  

  1          - Stokes theorem II                13.8
  2          - Divergence Theorem               13.5-8
  2          - Overview                         

        December 1: last day of class