 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,
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;
 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, September 19th to Wednesday, September 21th 2005.
More information.
 Section Types: Regular, Physics, BioChem flavors.
You will section online.
 Introductory Meeting: Tuesday, Sept 20 in Sci Center C at 8:30am.
 Lectures Start: Mon Sep 26 for MWF sections, and on
Tue Sep 27 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: Kory Byrns, byrns@fas.
 Texts: "Multivariable Calculus: Concepts and Contexts"
by James Stewart. We will use the third edition. This book is used by
all sections.
The Biochem section will additionally use: Y.A. Rozanovs book
"Probability theory, a concise course" 10 dollars).


All sections: Book by J. Stewart, ISBN 0534410049 
Bio section additionally uses the Book by Y.A. Rozanov, ISBN 0486635449 
 Weekly Recitations: the times will be arranged by Course Assistants (CAs)
 Question Center: 810 pm except Fridays and Saturdays in Loker Commons
Question Center Website
 Homework: Weekly HW assigned in parts each lecture.
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.
 Computers: The use of computers and other electronic aids
is not be permitted during exams. There is a Mathematica project,
which introduces you to an advanced computer algebra system.
 Exams:
First Hourly: Tuesday 18. October, 2005, Hall B, 78:30 PM
Second Hourly: Tuesday 15. November, 2005, Hall B, 78:30 PM
Final Examination: Saturday January 14'th, 2006
(according to printed course catalog)
 Grades:
First and second hourly 30 %
Homework 25 %
Mathematica project 5 %
Final exam 40 %

Final grade I 100 %
Final grade II Final exam grade

Actual grade: Max (I,II)
This grading policy is sometimes called "resurrection".
If your final grade is larger than the total grade I,
it is your course grade. So, theoretically, all options
are open.
We reserve the right to deny resurrection in cases which
indicate abuse like not turning in homework over long
periods of time or absence in class over extended
periods of time without excuse.
Experience shows that grade I and II are strongly correlated.
But resurrection can help students who have screwed
up a midterm due to time conflicts with other projects
or illness. It also can make a huge difference for students
who learn from midterm and homework mistakes, who work hard
during the last few weeks of class and during reading period.
 Calendar: 13 weeks
Su Mo Tu We Th Fr Sa  week special dates month
+
18 19 20 21 22 23 24  20. Intro meeting Sep
25 26 27 28 29 30 1  1 26/27 Start of lectures
2 3 4 5 6 7 8  2 Oct
9 10 11 12 13 14 15  3 10. Oct columbus day
16 17 18 19 20 21 22  4 18. Oct first midterm
23 24 25 26 27 28 29  5
30 31 1 2 3 4 5  6 Nov
6 7 8 9 10 11 12  7 11. Nov veterans day
13 14 15 16 17 18 19  8 15. Nov second midterm
20 21 22 23 24 25 26  9 2426. Thanksgiving
27 28 29 30 1 2 3  10 Dec
4 5 6 7 8 9 10  11
11 12 13 14 15 16 17  12
18 19 20 21 22 23 24  13
1 2 3 4 5 6 7  Reading period Jan
8 9 10 11 12 13 14  Reading period
15 16 17 18 19 20 21  Exam period
22 23 24 25 26 27 28 
+
 Day to day syllabus:
Hour Topic Book section Tue Thu
1. Geometry of Space 9/269/30
1  coordinates 9.1  1
 distance 
2  vectors 9.2 +
 dot product 9.3  2
3  cross product and planes 9.4 
2. Functions and Graphs 10/310/7
1  lines and planes 9.5 
 distance formulas  1
2  functions 9.6 +
graphs 
3  level curves  2
 quadrics 
3. Curves 10/1010/14
 Columbus day, no class
1  curves in space 10.1  1
 velocity 
 acceleration 10.2 
2  arc length 10.3 + 2
 curvature 10.4 
4. Surfaces 10/1710/21
1  review for first hourly  1
First Midterm (on chapters 910)
2  cylindrical coordinates 9.7  1
 spherical coordinates + 2
3  parametric surfaces 10.5  2
5. Functions 10/2410/28
1  functions 11.1  1
 continuity 11.2 
2  partial derivatives 11.3 +
Solutions to PDE's  2
3  linear approximation 11.4 
tangent planes
6. Gradient 10/3111/4
1  chain rule 11.5  1
implicit differentiation 
2  gradient 11.6 +
gradient and level curves  2
3  directional derivative 11.6 
direction of steepest decent 
7. Extrema 11/711/11
1  maxima, minima, saddle points 11.7  1
2  Lagrange multipliers 11.8 + 2
3  Veterans day (no class) 
8. Double Integrals 11/1411/18
1  review for second midterm  1
Second Midterm (through chapter 11.8)
2  double integrals 12.1/2  2
3  general regions 12.3 +
9. More double Integrals 11/2111/25
1  polar coordinates 12.4  1
 applications of double integrals 12.5 
2  surface area 12.6 
Thanksgiving break
10. Triple and line Integrals 11/2812/2
1  triple integrals 12.7  1
2  cylinder, spherical coordinates 12.8 +
3  vector fields 13.1 +
 line integrals 13.2  2
