 MathS21a: Multivariable Calculus, of the Harvard Summerschool 2006
 Instructor: Oliver Knill, SC434, (knill@math)
 Course assistant: Chris Phillips (phillips@fas)
 Lectures: Every Tuesday, Wednesday, and Thursday at 9:3011:00,
lectures start 9:30 sharp.
 Place: Lecture Hall Emerson 101 Google map, (marked local version).
 Sections: Thursday 89 Emerson Hall 307
12 PM, Emerson Hall 106
 Office hours: Oliver: Monday 15:3016:30, SC 434 and by appointment
 Website: http://www.courses.fas.harvard.edu/~maths21a/
 Text: Reading a textbook gives you a second opinion on the material.
We can recommend the new multivariable calculus book
by Brian Blank and Steven Krantz. It is available as a paperback edition
and is less than half the prize of other books like
"Multivariable Calculus: Concepts and Contexts" by James Stewart.
The BlankKrantz book fits this summerschool syllabus beautifully (evenso
we treat some topics in other order). Alternatively, any multivariable text
works. Homework will be distributed each class during lecture. Homework
problems are not given from textbooks.
 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.
 Quizes: We will have three short online multiple choice quizes.
The quizes do not enter the grade directly but allow to weight
the quiz score better. More details below.
 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
and all three quizes have to be completed. The project can
also be chosen by the student.
 Exams:
Two midterm exams and one final exam. The midterms on July 13 and July 27
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 20 sessions: 18 lectures plus 2 midterms
during 7 weeks from June 27, 2006 to August 10, 2006. This is
followed by a final examination week ending August 18. You might
want to check out the Official Summerschool Calendar
++ ++
Su Mo Tu We Th  Fr Sa Events WeekExamQuizProj
++ ++
25 26 27 28 29  30 1 June  1   *  
2 3 4 5 6  7 8 July  2    
9 10 11 12 13  14 15 13. hourly 3  *   
16 17 18 19 20  21 22  4   *  
23 24 25 26 27  28 29 27. hourly 5  *   
30 31 1 2 3  4 5 August  6   *  
6 7 8 9 10  11 12  7    * 
13 14 15 16 17  18 19 15. final   *   
++ ++
 Day to day syllabus:
1. Week: Geometry and Space
27. June: introduction, Eulidean space, vectors in the plane
28. June: vectors in space dot product, projection and component
29. July: cross product, lines, planes, distances, triple product
2. Week: Functions and Surfaces
4. July: independence day
5. July: functions, graphs, quadrics
6. July: implicit and parametric surfaces
3. Week: Curves and Partial Derivatives
11. July: curves, velocity, acceleration, chain rule
12. July: arclength, curvature, partial derivatives
13. July: first midterm (on week 12)
4. Week: Extrema and Lagrange Multipliers
18. July: gradient, linearization, tangents
19. July: extrema, second derivative test
20. July: extrema with constraints
5. Week: Double Integrals and Surface Integrals
25. July: double integrals, type I,II regions
26. July: polar coordinates, surface area
27. July: second midterm (on week 34)
6. Week: Triple Integrals and Line Integrals
1. August: triple integrals, cylindrical coordinates
2. August: spherical coordinates, vector fields
3. August: line integrals, fundamental thm of lineintegrals
7. Week: Exterior Derivatives and Integral Theorems
8. August: curl and Green theorem
9. August: curl and Stokes theorem
10. August: div and Gauss theorem
14 August: Final Review
15 August: Final exam (on week 17)
Exam: 1:30 PM Tuesday, August 15
