 Math21a: Multivariable Calculus.
 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,
computer graphics, 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;

 Prerequisites: Math 1b or equivalent
 Course change fees: All course change fees are waved
for students who change between Math 21a, Math 23a, Math 25a
and Math 55a until the 5'th Monday of the term.
 How to Sign Up: Input your time preferences
from Monday, Sep 18 to Wednesday, Sep 20 2006.
on the sectioning information page.
 Introductory Meeting: Tuesday, Sept 19, 2006, Sci Center C, 8:30am
 Lectures Start: Mon Sep 25 for MWF sections, and on
Tue Feb 26 for TTh sections
 Course Head: Oliver Knill
Science Center SC434
knill@math.harvard.edu
Office hours: Mon/Wed 2:304:00 and by appointment
 Head CA: Hannah Chung (hchung@fas)
 Text:
Multivariable calculus
by Brian Blank and Steven Krantz.
It is available as a paperback edition. The ISBN Number is
ISBN 1931914605
 Weekly Recitations: are organized and run by our Course Assistants (CAs)
 Question Center: The Math question center (MQC)
takes place in Science Center B10 this semester from 8:3010:30. Its start will be announced.
 Homework: Homework is assigned each lecture and due the next
lecture. No late homework is accepted. A fraction of the
HW score with weight of a week can be discarded and used as
"jokers", for example, in case of sickness. You are encouraged to
discuss solution strategies with classmates, your section leader or
your CA but you must write up answers yourself in your own words.
As with any academic work, external sources which were consulted should
be cited.
 Computers:
There is a Mathematica project, which introduces you to
an advanced and industrial strength computer algebra system.
The use of computers and other electronic aids is not
permitted during exams although.
 Exams:
First Hourly: Wednesday, Oct 18. 2006, Hall C, 78:30 PM
Second Hourly: Tuesday, Nov 14, 2006, Hall B, 78:30 PM
Final Examination: Monday Jan 22, Place/Time TBA
 Grades:
First and second hourly 30 %
Homework 25 %
Mathematica project 5 %
Final exam 40 %

Final grade 100 %
Calendar: 13 weeks
Su Mo Tu We Th Fr Sa  week special dates month
+
10 11 12 13 14 15 16 Sep 11: freshmen regist. Sep
17 18 19 20 21 22 23 Sep 19: intro meeting
24 25 26 27 28 29 30 1 Sep 25/26: lectures start
1 2 3 4 5 6 7 2 Oct
8 9 10 11 12 13 14 3 Oct 9: Columbus day
15 16 17 18 19 20 21 4 Oct 18: first hourly
22 23 24 25 26 27 28 5
29 30 31 1 2 3 4 6 Nov
5 6 7 8 9 10 11 7 Nov 10: Veteransday
12 13 14 15 16 17 18 8 Nov 14: second hourly
19 20 21 22 23 24 25 9 Nov 2326 thanksgiving
26 27 28 29 30 1 2 10 Dec
3 4 5 6 7 8 9 11
10 11 12 13 14 15 16 12
17 18 19 20 21 22 23 13 Dec 19: last day of class
24 25 26 27 28 29 30
31 1 2 3 4 5 6 Jan 212: reading period Jan
7 8 9 10 11 12 13
14 15 16 17 18 19 20 Jan 15: Martin Luther day
21 22 23 24 25 26 27 Jan 22: Final exam
+
 Day to day syllabus:
(*) There are a few major deviations from the book. These are
 partial differential equations, which we stress more
 parametric surfaces which we cover in more generality
 other coordinate systems are treated earlier
Otherwise, the syllabus matches quite well the book.
Hour Topic
1. Geometry of Space 9/259/29
1  coordinates 11.1
 distance, completion of square 11.2
2  vectors in space 11.2
 dot product 11.3
3  cross product and planes 11.4
2. Curves 10/210/6
1  lines and planes 11.5
 distance formulas (*)
2  curves in space 12.1
 velocity 12.2
 acceleration
3  arc length 12.3
3. Implicit surfaces 10/910/13
1  Columbus day, no class
2  curvature 12.4
functions and graphs 13.1
3  level surfaces 13.2
 quadrics
4 Parametric surfaces 10/1610/20
1  review for 1. hourly (week 13)
2  cylind./spher. coordinates 14.4
3  parametric surfaces (*)
5. Functions 10/2310/27
1  functions 13.3
 continuity
2  partial derivatives 13.4
Solutions to PDE's (*)
3  chain rule 13.5
implicit differentiation
6. Gradient 10/3011/3
1  gradient and relation 13.6
with level surfaces
2  directional derivative 13.6
direction of steepest decent
3  linear approximation 13.7
tangent planes
7. Extrema 11/611/10
1  maxima, minima, saddle points 13.8
2  Lagrange multipliers 13.9
3  Combined extremal problems
8. Double Integrals 11/1311/17
1  review for 2. hourly (week 47)
2  double integrals 14.1
3  general regions 14.2
polar coordinates 14.5
9. Triple integrals 11/2011/23
1  triple integrals 14.6
2  physical applications 14.7
 thanksgiving holiday
10. Line Integrals 11/2711/30
1  other coordinate systems 14.8
2  vector fields 15.1
3  line integrals 15.2
11. Greens theorem 12/412/8
1  fundamental thm line integrals 15.3
2  curl and divergence 15.4
3  Greens theorem 15.5
12. Stokes and Gauss 12/1112/15
1  flux integrals 15.6
2  Stokes theorem 15.7
3  Gauss theorem 15.8
 Overview integral theorems
13. Last lecture
1  Free lecture: i.e. review, catch up
Mathematica project due, last HW due
