 MathS21a: Multivariable Calculus, of the Harvard Summerschool 2004
 Instructor: Oliver Knill, SC434, knill@math
 Course assistant: Benjamin Bakker
bakker@fas
 Lectures: Every Tuesday, Wednesday, and Thursday at 9:3011:00,
lectures start 9:30 sharp.
 Place: Lecture Hall Emerson 101
 Sections: Thursday 89 and/or 12 PM, Lecture Hall Emerson 104
 Office hours: Oliver: Monday 10:0012:00, SC 434 and by appointment
Ben: Monday 19:0020:00 Math common room (near SC 434)
 Website: http://www.courses.fas.harvard.edu/~maths21a/
 Text:
Students who commit to follow this course will be handed out a preliminary
lecture notes package during the first week. Final versions of the notes,
handouts, class material and homework sheets will be distributed during
lectures. This allows to follow the course without a book. We recommend
however to read in a book like "Multivariable Calculus: Concepts and Contexts"
by James Stewart in addition to following the lectures.
 About this course:
 extends single variable calculus to higher dimensions;
 provides vocabulary for understanding fundamental
equations of nature like weather, planetary motion,
waves, heat, finance, epidemiology, or quantum mechanics.
 teaches important background needed for statistics,
computergraphics, bioinformatics, etc;
 provides tools for describing curves, surfaces, solids
and other geometrical objects in three dimensions;
 develops methods for solving optimization problems with
and without constraints;
 prepares you for further study in other fields of
mathematics and its applications;
 improves thinking skills, problem solving skills,
visualization skills as well as computing skills;
 Homework: Weekly HW will be assigned in three parts,
one in each lecture. You will receive a handout for
each problem set. Problems will not be assigned from books.
 Computers: The use of computers and other
electronic aids is not permitted during exams.
Mathematica projects are optional and will
teach you the basics of a computer algebra system.
Harvard has a site licence for Mathematica. Using
this software does not lead to any additional expenses.
The total time for doing the lab is 13 hours.
For people who prefer not to use any computers, there
will be the possibility to work on some challenge problems or
to write a little paper on your own. As experience has
shown, the later options require much more time resources
but they can be rewarding too. The project will be handed
in during the last lecture.
 Graduate Credit: This course can be taken for graduate credit.
The course work is the same. To fulfill the graduate credit,
a minimal 2/3 score has to be reached for the final project.
The project can also be chosen in the field of interests of
the student.
 Exams:
Two midterm exams and one final exam. The midterms
will be administered during class time in the usual lecture hall.
The final will take place during the examination period.
Online quizes are offered during nonexamination weeks 1,2,4,6.
They are "light", are of multiple choice nature and will help you to
gauge your progress in knowing the basic concepts. The score in
the quizzes will allow you to balance the scores in the midterms.
 Grades:
First and second hourly 40 % weighted according to quizz
Homework 25 %
Project 5 %
Final 30 %
The higherscoring midterm will be worth 20+X%, the other midterm
will be worth 20X%, where X is the total score in the online quizzes
scaled from 0 to 8. If the grade on the final exam is higher than
the grade from the composite score, then the final grade for the
course will be equal to the grade on the final exam. Active class
participation and attendence can boost your final grade by up to 5%.
 Calendar: we have 21 sessions: 19 lectures plus 2 midterms
during 7 weeks from June 28, 2004 to August 12, 2004. This is
followed by a final examination week ending August 20. You might
want to check out the Official Summerschool Calendar
++ ++
Su Mo Tu We Th  Fr Sa Events WeekExamQuizProj
++ ++
27 28 29 30 1  2 3 June  1   *  
4 5 6 7 8  9 10 July  2   *  
11 12 13 14 15  16 17 15. hourly 3  *   
18 19 20 21 22  23 24  4   *  
25 26 27 28 29  30 31 29. hourly 5  *   
1 2 3 4 5  6 7 August  6   *  
8 9 10 11 12  13 14  7    * 
15 16 17 18 19  20 24 17. final   *   
++ ++
 Day to day syllabus:
1. Week: Geometry and Space
29. June: introduction, space, coordinates, distance
30. June: vectors, dot product, projections
1. July: cross product, lines
2. Week: Functions and Surfaces
6. July: planes, distance formulas
7. July: functions, graphs, quadrics
8. July: implicit and parametric surfaces
3. Week: Curves and Partial Derivatives
13. July: curves, velocity, acceleration, chain rule
14. July: arclength, curvature, partial derivatives
15. July: first midterm (on week 12)
4. Week: Extrema and Lagrange Multipliers
20. July: gradient, linearization, tangents
21. July: extrema, second derivative test
22. July: extrema with constraints
5. Week: Double Integrals and Surface Integrals
27. July: double integrals, type I,II regions
28. July: polar coordinates, surface area
29. July: second midterm (on week 34)
6. Week: Triple Integrals and Line Integrals
3. August: triple integrals, cylindrical coordiantes
4. August: spherical coordinates, vector fields
5. August: line integrals, fundamental thm of lineintegrals
7. Week: Exterior Derivatives and Integral Theorems
10. August: curl and Green theorem
11. August: curl and Stokes theorem
12. August: div and Gauss theorem
17. August: Final exam (on week 17)
1:30 PM.
