 MathS21a: Multivariable Calculus
 Instructors: Daniel Goroff and Oliver Knill
 Course Assistants: Jonathan Kaplan (jkaplan@math.harvard.edu) and
Izzet Coskun (coskun@math.harvard.edu)
 Office hours:
Wednesday 1112:30 for Goroff in SC427
Mondays 1112:30 for Knill in SC434
Tuesdays 1314:00 for Kaplan in SC428b
 Website: http://www.courses.fas.harvard.edu/~maths21a/
 Time: Every Tuesday, Wednesday, and Thursday 9:3011 except 7/4
The lectures start at 9:30 sharp.
 Place: Emerson 101 (beginning Thursday 6/27/2002)
 Problem sessions: Thursdays 12 in SC110
Thursdays 89 in SC112
 Homework: Weekly HW assigned in three parts, one in each lecture.
All three parts are due Tuesdays at the start of class.
No late homework is accepted. You are encouraged to
discuss solution strategies with classmates, but you
must write up answers yourself in your own words. As
with any academic work, please cite sources consulted.
 Project: As an optional 8th homework, you may choose to hand in:
 a computer laboratory project
 challenge problems you have completed
 certain projects and problems from the book
If you do decide to work on an 8th assignment, only the best 7 of
your 8 HW scores will count. Projects are due on 8/12 at 9:30 am.
Graduate students are expected to do 2/3 of the 8'th homework to get
credit for this course.
 Text: James Stewart, "Multivariable Calculus, Concepts and
Contexts" can be purchased on line or at the Harvard Coop. Student
solution manuals are also available. Please read relevant
sections of the textbook before you come to class.
 Calendar: (18 lectures in 7 weeks)
++
Su Mo Tu We Th  Fr Sa
 
23 24 25 26 27 28 29 June 1
30 1 2 3   5 6 July 2
7 8 9 10 11 12 13 3
14 15 16 17 18 19 20 4
21 22 23 24 25 26 27 5
28 29 30 31 1  2 3 August 6
4 5 6 7 8  9 10 7
11 12 14 16 17 8
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 Exams: Two Midterms
Thursday 7/11 at 9:30, in usual room.
Thursday 7/25 at 9:30, in usual room.
One cumulative Final Examination on Wednesday 8/14 at 9:15am.
These three exams are required. Please note the dates and times.
In general, we expect roughly a third of each test to consist
of True/False or Conceptional Questions directly from the text,
and another third to consist of assigned or unassigned
exercices directly from the book.
 Grades: There is a total target of T=1000 points for this course.
Roughly, we expect people who end up with over 900 points to
receive some kind of A, people who have 800900 points to receive
some kind of B, etc. A maximum of 820 points can be obtained
during the term:
7 Homeworks, each 60 points 420 points
2 Midterms, each 200 points 400 points
If you earn N points during the term, your final exam will be worth
a possible TN points. For example,
1) If you enter the final with all 820 points and get
50% correct on the final, your course total will be:
820 + (1000820) * 50/100 = 910 (probably a low A or high B).
2) If you enter the final with 430 points and get
70% correct on the final, your course total will be:
430 + (1000430) * 70/100 = 839 (probably a low B).
Notice this means you can earn back on the final any point you
miss during the term, so you always have an incentive to keep
working. This grading system is very kind to anyone who
eventually masters the material by the day of the final, but very
risky for anyone who tries to wait until then. Please keep up,
work lots of problems, go over everything you did not get right
the first time, ask lots of questions, and you will do fine.
 About this course:
 extends single variable calculus to higher dimensions;
 provides vocabulary for understanding the fundamental
equations of nature (e.g., weather, heat, planetary
motion, waves, profit maximization, quantum world);
 provides tools for describing curves, surfaces, and other
graphical objects in three dimensions;
 develops methods for solving optimization problems with and
without constraints;
 prepares you for further study in many other fields of
mathematics and its applications;
 improves thinking skills, problem solving skills,
visualization skills, and computing skills.
 Syllabus:
Date Topic Book section
I) Geometry of Space
6/25 Tue  coordinates 9.1
 distance
6/26 Wed  vectors 9.2
 dot product 9.3
6/27 Thu  cross product 9.4
 lines and planes 9.5
II) Functions and Graphs
7/2 Tue  functions, graphs 9.6
 level curves, quadrics
7/3 Wed  cylindrical coordinates 9.7
 spherical coordinates
7/4 Thu  holiday
III) Curves and Surfaces
7/9 Tue  curves in space 10.1
 velocity,acceleration 10.2
7/10 Wed  arc length, curvature 10.3
parametric surfaces 10.5
7/11 Thu  First Midterm (chapters 910)
IV) Partial Derivatives
7/16 Tue  functions 11.1
 continuity 11.2
 partial derivatives 11.3
 linear approximation 11.4
7/17 Wed  chain rule 11.5
 gradient, directional deriv. 11.6
7/18 Thu  maxima, minima, saddle points 11.7
 Lagrange multipliers 11.8
V) Multiple Integrals
7/23 Tue  double integrals 12.1
 iterated integrals 12.2
 general regions 12.3
 polar coordinates 12.4
7/24 Wed  surface area 12.6
 triple integrals 12.7
 polar, spherical coordinates 12.8
 change of variables 12.9
7/25 Thu  Second Midterm (until chapter 12)
VI) Line Integrals
7/30 Tue  vector fields 13.1
 gradient fields
7/31 Wed  line integrals 13.2
 fundamental thm line integrals 13.3
8/1 Thu  application: work
application: electrostatics
VII) Integral Theorems
8/6 Tue  Greens theorem 13.4
 curl and divergence 13.5
8/7 Wed  surface integrals 13.6
 Stokes theorem 13.7
8/8 Thu  Gauss theorem 13.8
 Applications
8/14 Wed Final Examination 9:15 (chapters 913)
