Course name

Harvard Summer School course Maths 21a, Multivariable Calculus, CRN 30189, Summer 2011


Oliver Knill,, SC 432, Harvard University

Course Assistant

Alex Isakov,, Harvard University

Meeting time

Tuesday/Thursday 8:30 - 11:30 in Hall E, and a weekly problem section on Thursday afternoon in a room to be arranged. We start at 8:30 sharp. There is a break after 90 minutes.

Problem sessions

Additional to the lectures there is a problem session on Thursday 1-2 PM in SC 507. it is highly recommended to participate in these sections.


A solid background in single variable calculus is required. If you want to look some old things up, here is some material from a single variable course I taught last spring.

The course

It extends single variable calculus to higher dimensions. You will see that the structures are much richer and that the fundamental theorem of calculus comes in two flavors in dimension 2 and has three incarnations in dimension 3.
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.
It develops methods for solving problems. Examples are optimization problems with and without constraints, 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. No programming experience is required however.
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 or number theory, where similar structures and problems appear.
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 insight you gain this summer will remain true and relevant forever because mathematical truth is eternal.

The workload

For this standard multivariable calculus course the total lecture time is roughly the same as for our standard Math21a course in the Harvard semester. We cover the same material but do this slightly more compressed and streamlined. This course is a "tough" course. It requires about 20 hours of work per week in average from a student including lectures, recitation, reading, homework and exam preparation. Taking two heavy classes simultaneously leads to a 40 hour work week. It is possible, but very demanding. Taking two courses only leaves time for enjoying the summer in Cambridge if the work time is well managed. If you work, have a heavy sports or music schedule or want to reserve time for a personal project, I recommend you take this course without a second course.


Every Tuesday and Thursday at 8:30-11:30, Science Center Lecture Hall E The course time is condensed to 6 weeks with 6 hours of lecture every week. The 3 hour lectures can be a bit tough. You can bring a snack since we have a short break after 90 minutes. I divided the material in 24 units. Two units of 90 minutes each are taught each Tuesday and Thursday. The classes in which the exam takes place are a bit more condensed.

Office hours

Oliver: Monday 15:30-17:00, SC 432 and by appointment Course assistant: Alex, Office hours Alex: Thursday 2 PM.


All course material and homework will be handed out in class. In principle, you do not need a textbook. Reading a textbook can give you a "second opinion" on the material. A widely used textbook is "Multivariable Calculus: Concepts and Contexts" by James Stewart. but any multivariable text works; they are all very similar to each other. Especially, any old edition does fine. We cover the material which can be found in chapters 9-12 of Stewart or chapters 10-14 in Varberg Purcell Rigdon or chapters 10-15 in Smith Minton.


Homework problems are included on the lecture notes. Each lecture unit has 5 problems so that you do 20 problems each week. Homework will be returned weekly. Homework is due on Tuesdays except for the last week, where the homework is due Tuesday and Thursday. It is strongly recommended that you do the first 10 problems after the first day. This will also make it more easier to make sense the second day each week.


There are two midterm exams and one final exam. The midterms are administered during class time in the usual lecture hall E. The final exam will take place during the examination period, again in the usual lecture hall. Due to the summer school schedule you will naturally not have much time to prepare for the exams. It will therefore be important to keep always up with the material and ask if something is not clear.


First and second hourly     40 % total
Homework                    25 %
Project                      5 %
Final                       30 %
Active class participation and attendance can boost your final grade by up to 5%.

Graduate Credit

This course can be taken for graduate credit. The course work is the same. To fulfill the graduate credit requirements, a minimal 2/3 score must be reached for the final Mathematica project.

Mathematica project

The use of computers and other electronic aids can not permitted during exams. A Mathematica project will teach you the basics of a computer algebra system Mathematica 8.0 for which Harvard has a site license. This software does not lead to any additional expenses and the total time for doing the lab is 3-4 hours. The project will be handed in at the beginning of the August 2 lecture.


So Mo Tu We Th Fr Sa   Week  Special Events
26 27 28 29 30  1  2   1     Jun 28  First class
 3  4  5  6  7  8  9   2
10 11 12 13 14 15 16   3     Jul 14  First midterm
17 18 19 20 21 22 23   4
24 25 26 27 28 29 30   5     Jul 28  Second Midterm
31  1  2  3  4  5  6   6     Aug  2  Mathematica due
 7  8  9 10 11 12 13   7     Aug 11  Final exam

Day to day lecture

1. Week:  Geometry and Space

   Lect 1-2   6/28 Space, Vectors, Dot Product
   Lect 3-4   6/30 Cross product, Lines/Planes, Distances

2. Week:  Surfaces and Curves

   Lect 5-6   7/5  Implicit and Parametric Surfaces
   Lect 7-8   7/7  Curves, Chain Rule, Arc Length

3. Week:  Gradient and Linearization

   Lect 9-10  7/12 Partial Derivatives, Review
   Lect 11-12 7/14 First hourly. Linearization

4. Week:  Extrema and Double Integrals

   Lect 13-14 7/19 Extrema and Lagrange
   Lect 15-16 7/21 Double integrals and Surface area

5. Week:  Multiple Integrals and Line Integrals

   Lect 17-18 7/26 Triple integrals also in Spherical coordinates
   Lect 19-20 7/28 Second hourly. Line integrals

6. Week:  Vector Fields and Integral Theorems

   Lect 21-22 8/2  Curl, Greens theorem, Flux
   Lect 23-24 8/4  Stokes and Divergence theorem

7. Week:  Review and Final exam (Aug 11)