 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, Jan. 30 to Wednesday, Feb 1 2006.
on the sectioning information page.
 Section Types: Regular, Physics, BioChem flavors.
You will section online.
 Introductory Meeting: Wednesday Feb 1 in Sci Center C at 8:30am.
 Lectures Start: Mon Feb 6 for MWF sections, and on
Tue Feb 7 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:
 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 March 14. 2006, Hall D, 78:30 PM
Second Hourly: Tuesday April 18, 2006, Hall B, 78:30 PM
Final Examination: Tuesday, May 23, 2006
 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
+
Su Mo Tu We Th Fr Sa Week
1 2 3 4 Feb
5 6 7 8 9 10 11 1 6/7 start of lectures
12 13 14 15 16 17 18 2
19 20 21 22 23 24 25 3 20.Feb Presidents day
26 27 28 1 2 3 4 4 Mar
5 6 7 8 9 10 11 5
12 13 14 15 16 17 18 6 March 14. First hourly
19 20 21 22 23 24 25 7
26 27 28 29 30 31 1 Spring recess Apr
2 3 4 5 6 7 8 8
9 10 11 12 13 14 15 9
16 17 18 19 20 21 22 10 April 18. Second hourly
23 24 25 26 27 28 29 11
30 1 2 3 4 5 6 12 Mai
7 8 9 10 11 12 13 Reading period
14 15 16 17 18 19 20 17. end reading period
21 22 23 24 25 26 27 26. End exam period
+
 Day to day syllabus:
Hour Topic Book section Tue Thu
1. Geometry of Space 2/62/11
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 2/132/18
1  lines and planes 9.5 
 distance formulas  1
2  functions 9.6 +
graphs 
3  level curves  2
 quadrics 
3. Curves 2/202/25
 Presidents 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 2/273/3
1  cylindrical coordinates 9.7  1
 spherical coordinates + 2
2  parametric surfaces 10.5  2
3  functions 11.1  1
 continuity 11.2 
5. Functions 3/63/10
1  partial derivatives 11.3 +
Solutions to PDE's  2
2  linear approximation 11.4 
tangent planes
3  chain rule 11.5  1
implicit differentiation 
6. Gradient 3/133/17
1  review for first hourly  1
2  gradient 11.6 +
gradient and level curves  2
3  directional derivative 11.6 
direction of steepest decent 
7. Extrema 3/203/24
1  maxima, minima, saddle points 11.7  1
2  Lagrange multipliers 11.8 + 2
3  Combined problems 11.8  2
Spring recess
8. Double Integrals 4/34/7
1  double integrals 12.1/2  2
2  general regions 12.3 +
3  polar coordinates 12.4  1
9. Surface Area 4/104/14
1  surface area 12.6 
2  triple integrals 12.7 + 1
3  cylinder, spherical coordinates 12.8  1
10. Line Integrals 4/174/21
1  review for second hourly 
2  vector fields 13.1 +
3  line integrals 13.2  2
