Math 21 a Math21a, Fall 2001
Course Head: Prof. Clifford H. Taubes

Mainpage Syllabus Calendar Homework Exams Supplements Comput. assigns Links

Math 21a Fall 2001 Course Announcement

The subject: Since Math 21a is a mathematics course, I will start with a digression to introduce a definition for the science of mathematics so as to distinguish it from the other sciences:
Mathematics consists of the study of all imaginable worlds with the goal of uncovering transcendent, universal relationships and underlying symmetries.
By way of contrast, fields such as physics, chemistry or biology, or even the more social sciences such as economics are concerned with the details of the particular universe that we inhabit. This is to say that the charge for the other sciences can be summarized as follows:
Provide a predictive understanding of the given universe.
The other sciences find mathematics useful, and often remarkably so, because the underlying relationships which transcend our particular context yield predictions which would be unfathomable with a focus that restricts solely to our own world.
With the digression now over and with the preceding remarks as background, consider that Math 21a (and Math 21b as well) focuses on various concrete and abstract properties of 2 and 3 dimensional space. Of particular concern here in Math 21a are functions whose values vary over such spaces - this is to say that the subject is calculus for functions of two or more variables. Such functions arise ubiquitously in the sciences. Here are some examples: First, a biologist might be interested in the concentration of a certain cellular protein as a function of the age of the cell and the distance from the cell's nucleus. In this case, the function measures protein concentration and it depends on the two variables, age and distance. Second, a physicist might be concerned with the strength of a current in a superconductor as a function of the temperature, the strength of the ambient magnetic field and the density of impurities. So here, the function in question depends on the three variables, temperature, magnetic field strength and impurity density. Third, an environmental scientist might be concerned with the concentration of mercury in the Charles River as a function of distance from Boston Harbor and water flow rate. Here, the interest is in a function of two variables, distance and flow rate. Meanwhile, a geologist might worry about a function that gives the height above sea level of the Modoc Plateau in California as a function of the three variables, latitude, longitude and time (measured in units of millions of years). Finally, an economist might be concerned with the Dow Jones average as a function of various other economic indicators.
In any event, our concern in Math 21a is not so much on examples, but on developing various general techniques from calculus for the purpose of analyzing and predicting the behavior of functions of two or more variables. None-the-less, concrete examples abound in the course. However, there are discussions ahead that can seem quite abstract; but be aware that even the most abstract subjects in this course have applications in the other sciences, even if such applications are not mentioned here.
The sections: The precise subjects that are covered in Math 21a depend to some extent on the particular type of section that you choose. In this regard, you should know that Math 21a is taught in small sections of size 25 students, with sections labeled as "Regular", "Physics", "BioChem" and "Computer Science". A section without a special label is automatically a Regular section.
  • All sections cover at least the following topics: Functions of several variables, differentiation and integration of functions of several variables, parametric curves and surfaces, optimization of functions, vector fields, linear approximations and various topics in partial differential equations.
  • The Regular and Physics sections also cover the following: Line and surface integrals, Green's theorem, the divergence theorem and Stokes' theorem. These are the multi- variable generalizations of the Fundamental Theorem of Calculus.
  • The Physics sections will use examples drawn from Physics to illustrate some of the covered topics. Certain of the homework problems assigned to this section will also come from Physics.
  • The BioChem sections covers, in addition to the material in the first point, various introductory topics in statistics and probability. Note that no particular background is needed in either biology or chemistry in this section. Thus, the material is accessible to all.
  • The Computer Science section covers, in addition to the material in the first point, various introductory topics in logic, probability and combinatorics. In spite of its name, no particular background in Computer Science is needed for this section.
Detailed syllabi for the various types of sections are provided below.
Which kind of section should you choose?
  • If you contemplate being a physics concentrator, you would benefit by being in the Physics section.
  • In any event, if you are planning to take either the Physics 15-16 sequence or Physics 11 and other physics or Applied Science courses, you should enroll in either a Regular or a Physics section.
  • If you are planning to concentrate in BioChemical Sciences, then you are strongly urged by that department to enroll in a BioChem section.
  • If you plan to major in economics or other social sciences, the BioChem section might be your best choice.
  • If you plan to major in Computer Science, then you should find the Computer Science section to be the most relevant.
  • No matter what, you can't go wrong in a Regular section so if you aren't sure of your major, take a Regular section.
To section: Use any Harvard computer to telnet to 'hilbert.math.harvard.edu'. When prompted to 'login', type 'section'. At the password prompt, press 'enter'.
telnet to hilbert.math.harvard.edu
Follow the online instructions from here. Alternately, from any web browser, go to the Math Department's home page, http://www.math.harvard.edu, and click on the "sectioning" link on the upper right corner of the page. This done, follow the instructions. If there is a problem with your section assignment, contact Susan Milano in office 308 of the science center of via email at milano@math.harvard.edu).
Course Head: Clifford Taubes, Science Center 504, email chtaubes@math.harvard.edu . Drop in office hours on Mondays 9:30-11 and Fridays 2-3:30.
Prerequisites: Math 1b with a satisfactory grade, or AB-BC score of at least 4, or scores of at least 20, 8, 4 on the respective three Harvard University Math Placement Tests.
Textbooks: All of the sections require Multivariable Calculus by Ostebee and Zorn with the Student Solutions Manual, published by Saunders College Publishing. The BioChem sections also require Fundamentals of Biostatistics by Rosner, published by Duxbury Press. The Computer Science section requires a special course pack that reproduces Chapters 13 and 14 from Calculus, Vol II by Apostol. These books are available at the Harvard Coop exept for the Computer Science section's course pack, which can be obtained in the basement of the Science Center. In this regard, just ask for the Math 21a Computer Science section course pack.
Class meetings and problem sessions: The first class meeting, which everyone should attend, is on Thursday, September 13 at 8am in Science Center lecture hall C. Except for this one meeting and for the course wide exams, you meet in your assigned section. The section meets for a total of three hours per week, either one hour each on Mondays, Wednesdays and Fridays, or for one and one half hours each on Tuesdays and Thursdays. Each student is also assigned to a 1-hour math problem session, conducted weekly by a course assistant. The meeting time for the problem session will be arranged in your section during the first week of classes. You may attend more than one problem session per week; and the schedule of all problem sessions will be posted on the Math 21a website and on the Calculus Office bulletin board outside of Science Center 308.
The first meeting of MWF sections is on Monday, September 17 at the posted time. The first meeting for the TTh sections is on Tuesday, September 18.
Homework: A substantial problem set will be assigned once each week to be turned in as instructed in the subsequent week. You are strongly encouraged to discuss the homework with your fellow students and to form study groups to work these assignments. However, you must write up the solutions by yourself, and you must note the names of your coworkers somewhere on the homework. (This last point is simply a matter of professional ethics.) The lowest homework score will be disregarded when your average homework grade is computed.
The weekly homework assignments will be posted on the Math 21a web site. The answers to the homework assignments will appear after the due date on the web site as well. Moreover, selected problems from the homework will be discussed in the problem sessions. Homework assignments that are submitted after their assigned due date will be accepted at my discretion. In any event, no more than two late homework assignments will be accepted per student over the course of the semester.
In addition to the weekly homework assignment, various problems of a more routine sort will be suggested for the subsequent class meeting. These are not to be turned as their answers are either in the Student Solutions Manual or will be provided otherwise. However, you are strongly urged to work them on your own or with others in the class because their purpose is to supply practice with the techniques and ideas that are presented in the lectures. By the way, you are also strongly encouraged to try on your own other problems from the text to hone your ability with the concepts and techniques. Don't feel that you should limit yourself to the suggested problems. In this regard, note that the Student Solution Manual answers most of the odd numbered problems in the Ostebee and Zorn book.
Computer assignments: There will be three specially designated assignments during the semester whose purpose is to introduce you to graphing and mathematical manipulation computer programs. The use of computer technology to solve mathematical problems is one of the great advances of our age, and so I want all of you to have at least a fleeting introduction to this side of the subject. No prior knowledge of the relevant software technology is required to work these assignments. More details will come later in the semester.
Exams: There are two course-wide "midterms" and a final. The first midterm will take place on Wednesday, October 10 from 7-9pm in Science Center lecture halls C and D, and the second on Wednesday, November 14, from 7:30-9:30pm in Science Center C & E. Please note that the times and rooms differ for the two midterms. In any event, be sure to mark these dates on your calendar now, as no make-ups will be given. The final exam is scheduled by the University for a date in January. According to the Course Catalogue, the preliminary schedule has the final on Tuesday, January 22. The University will confirm this date later in the semester.
Grading: Your final grade will be based on your performance on the homework (30%), the computer assignments (3%), the two midterms (10% for the first and 15% for the second), and the final (42%). A small upward adjustment in the grade is possible when the final is dramatically better than the average of the midterms and the homework.
Computers and calculators: The visualization of surfaces and other geometric phenomena is an important aspect of this course. In as much as computerized graphing programs aid you to develop this ability, you are encouraged to employ them as part of the learning process. In this regard, the scheduled computer assignments are designed to introduce you to the tool of computer graphing and mathematical manipulation. However, be forewarned that for the purposes of this course, computers should be considered solely as an aid to the development of geometric intuition. This course is teaching various concepts whose applications may or may not be facilitated by a computer. However, without a strong understanding of the underlying concepts, the computer won't be much use. The point here is that computers can do many things, but they can't yet think for you. As powerful as today's computers are (and tomorrow's will be), none of us will live to see the day when they can turn pig feed into gold. In any event, the use of computers and other electronic aids will not be permitted during exams. (Bring only your brain and some pencils.) With this in mind, note that various homework problems ask you to sketch or otherwise describe various geometric objects. With the exception of the specially designated computer assignments, you are strongly advised to struggle with these first without electronic aids, as they may be quite trivial with a graphing program.
Words of Caution and Advice: This course will be more demanding then your previous mathematics courses at Harvard and elsewhere. In particular, the assignments will be time consuming and you should plan now to set aside regular hours to wrestle with them. It is virtually impossible to do well in this course without working the homework assignments in a timely fashion. Note also that this course is fast paced, and new material builds on old. Thus, do not fall behind. If you find yourself falling behind, please contact your section's teacher immediately to discuss options for personal help. Indeed, Harvard provides many services along these lines for its students, and your section teacher can help you find them.
When you are working your assignments, keep in mind that your success in this course will require more than just memorizing formulas and "plugging in values". Numerical calculations are still important, but play a smaller role than in 1-variable calculus. Here is the key to success:
Understand the underlying concepts and then work enough problems so that you can employ them in any example thrown at you.
(In this regard, you will consistently battle with homework and exam problems that differ significantly from material discussed in class.)

Math 21a Fall 2000 Syllabus Outline

Regular Sections
Textbook: "Multivariable Calculus with the Student Solutions Manual" by Ostebee and Zorn,

  1. Section 1.1: 3-d space and surface.
  2. Section 1.2: Parametric curves.
  3. Section 1.3-1.4: Vectors, vector valued functions.
  4. Section 1.5: Derivatives, anti-derivatives and motion.
  5. Sections 1.6-1.7: Dot product, lines and planes
  6. Sections 1.7-1.8: Lines, planes and cross product.
  7. Appendix C: Introduction to matrices.
  8. Sections 2.1 & 1.1: Functions of several variables.
  9. Sections 2.2-2.3: Partial derivatives, contours & linear approximations.
  10. Sections 2.3-2.4: Linear approximations and the gradient.
  11. Sections 2.5-2.6: Derivative theory, high order derivatives, quadratic approx.
  12. Section 2.7: Max-min, critical points and extreme points.
  13. Sections 2.7 & 4.4: More on extreme points, constrained optimization.
  14. Section 4.4: Constrained opimization.
  15. Section 2.8: Multivariable chain rule.
  16. Section 3.1: Multiple integrals.
  17. Section 3.2: Calculating integrals by iteration.
  18. Section 3.3: Integrals in polar coordinates.
  19. Section 3.4: Integrals in spherical and cylindrical coordinates.
  20. Section 5.1: Line integrals and vector fields.
  21. Section 5.2: The fundamental theorem of line integrals.
  22. Section 5.3: Line integrals and Green's theorem.
  23. Section 5.4: Surfaces and parameterizations.
  24. Section 5.5: Surface integrals
  25. Section 5.6: Derivatives of vector valued functions, div and curl.
  26. Section 5.7: Stoke's theorem.
  27. Section 5.7: Divergence theorem.
  28. Section 5.7: More on Stoke's & Divergence Theorems.
  29. Handouts: Differential equations.
  • Three computer assignments.
  • Two "midterm" exams: Wed., Oct. 10 from 7-9pm in Science Center C & D, and Wed., Nov. 14 from 7:30-9:30 in Science Center C & E.
  • A final exam scheduled by the University; the preliminary date is Tue, Jan. 22.
Physics Sections
Textbook: "Multivariable Calculus with the Student Solutions Manual" by Ostebee and Zorn

  1. Section 1.1: 3-d space and surface.
  2. Section 1.2: Parametric curves.
  3. Section 1.3-1.4: Vectors, vector valued functions.
  4. Section 1.4-1.5: Derivatives, anti-derivatives and motion.
  5. Sections 1.6-1.7: Dot product, lines and planes
  6. Sections 1.7-1.8: Lines, planes and cross product.
  7. Sections 1.1, 2.1-2.2: Functions of several variables, partial derivative, gradients.
  8. Section 5.1: Line integrals and vector fields.
  9. Section 5.2: The fundamental theorem of line integrals.
  10. Section 2.3 Linear approximations
  11. Section 2.4: The gradient and directional derivatives.
  12. Appendix C: Introduction to matrices.
  13. Sections 2.5-2.6: Derivative theory, high order derivatives, quadratic approx.
  14. Section 2.7: Max-min, critical points and extreme points.
  15. Sections 2.7 & 4.4: More on extreme points, constrained optimization.
  16. Section 4.4: Constrained opimization.
  17. Section 2.8: Multivariable chain rule.
  18. Section 3.1: Multiple integrals.
  19. Section 3.2: Calculating integrals by iteration.
  20. Section 3.3: Integrals in polar coordinates.
  21. Section 3.4: Integrals in spherical and cylindrical coordinates.
  22. Section 5.3: Line integrals and Green's theorem.
  23. Section 5.4: Surfaces and parameterizations.
  24. Section 5.5: Derivatives of vector valued functions, div and curl.
  25. Section 5.7: Stoke's theorem.
  26. Section 5.7: Divergence theorem.
  27. Section 5.7: More on Stoke's & Divergence Theorems.
  28. Handouts: Differential equations.
  • Supplementary handouts: Work & Energy, Angular Momentum, Planetary Motion, Relativity, Center of Mass, Electricity & Magnetism, Euler's Equation, Wave Equations, Heat Equation, Laplace's Equation, Schrodinger's Equation, Dirac's Equation.
  • Three computer assignments.
  • Two "midterm" exams: Wed., Oct. 10 from 7-9pm in Science Center C & D, and Wed., Nov. 14 from 7:30-9:30 in Science Center C & E.
  • A final exam scheduled by the University; the preliminary date is Tue, Jan. 22.

BioChem Sections
Textbooks:
  • "Multivariable Calculus with the Student Solutions Manual" by Ostebee and Zorn
  • "Fundamentals of Biostatistcs" by Rosner.

  1. Section 1.1: 3-d space and surface.
  2. Section 1.2: Parametric curves.
  3. Section 1.3-1.4: Vectors, vector valued functions.
  4. Section 1.5: Derivatives, anti-derivatives and motion.
  5. Sections 1.6-1.7: Dot product, lines and planes
  6. Sections 1.7-1.8: Lines, planes and cross product.
  7. Appendix C: Introduction to matrices.
  8. Sections 2.1 & 1.1: Functions of several variables.
  9. Sections 2.2-2.3: Partial derivatives, contours & linear approximations.
  10. Sections 2.3-2.4: Linear approximations and the gradient.
  11. Sections 2.5-2.6: Derivative theory, high order derivatives, quadratic approx.
  12. Section 2.7: Max-min, critical points and extreme points.
  13. Sections 2.7 & 4.4: More on extreme points, constrained optimization.
  14. Section 4.4: Constrained opimization.
  15. Section 2.8: Multivariable chain rule.
  16. Section 3.1: Multiple integrals.
  17. Section 3.2: Calculating integrals by iteration.
  18. Section 3.3: Integrals in polar coordinates.
  19. Section 3.4: Integrals in spherical and cylindrical coordinates.
  20. Rosner Chapter 2: Descriptive statistics.
  21. Rosner Chapter 3: Beginning probability; addition laws, multiplication laws, conditional probability, total probability, Bayes' theorem.
  22. Rosner Chapter 4: Discrete probability distributions; binomial distributions, Poisson distributions.
  23. Rosner Chapter 5: Continuous probability; density functions, mean and variance, normal distributions, approximation by normal distributions.
  24. Handouts: Differential equations.
  • Three computer assignments.
  • Two "midterm" exams: Wed., Oct. 10 from 7-9pm in Science Center C & D, and Wed., Nov. 14 from 7:30-9:30 in Science Center C & E.
  • A final exam scheduled by the University; the preliminary date is Tue, Jan. 22.


Computer Science Sections
Textbooks:
  • "Multivariable Calculus with the Student Solutions Manual" by Ostebee and Zorn
  • and the Math 21a Computer Science Section course pack that contains Chapters 13 and 14 from Calculus, Vol. II by Apostol.

  1. Section 1.1: 3-d space and surface.
  2. Section 1.2: Parametric curves.
  3. Section 1.3-1.4: Vectors, vector valued functions.
  4. Section 1.5: Derivatives, anti-derivatives and motion.
  5. Sections 1.6-1.7: Dot product, lines and planes
  6. Sections 1.7-1.8: Lines, planes and cross product.
  7. Appendix C: Introduction to matrices.
  8. Sections 2.1 & 1.1: Functions of several variables.
  9. Sections 2.2-2.3: Partial derivatives, contours & linear approximations.
  10. Sections 2.3-2.4: Linear approximations and the gradient.
  11. Sections 2.5-2.6: Derivative theory, high order derivatives, quadratic approx.
  12. Section 2.7: Max-min, critical points and extreme points.
  13. Sections 2.7 & 4.4: More on extreme points, constrained optimization.
  14. Section 4.4: Constrained opimization.
  15. Section 2.8: Multivariable chain rule.
  16. Section 3.1: Multiple integrals.
  17. Section 3.2: Calculating integrals by iteration.
  18. Section 3.3: Integrals in polar coordinates.
  19. Section 3.4: Integrals in spherical and cylindrical coordinates.
  20. Apostol 13.1-13.9: Set theory, Boolean algebra, and probability.
  21. Apostol 13.10-13.14: Combinatorial analysis, conditional probability.
  22. Apostol 13.15-13.23: Bernoulli trials, countably infinite sample spaces, time spent in "while loops".
  23. Apostol 14.1-14.10: Random variables, discrete and continuous distributions, random number generators.
  24. Apostol 14.11-14.22: Two-dimensional discrete and continuous distributions, Bayes' Rule.
  25. Apostol 14.23-14.26: Calculating expectation and variance by multiple integration, functions of random variables.
  26. Handouts: Differential equations.
  • Supplementary notes on Boolean algebra and bit vectors, Bernoulli trials and while loops, Congruential random number generators, Using recursion in counting.
  • Three computer assignments.
  • Two "midterm" exams: Wed., Oct. 10 from 7-9pm in Science Center C & D, and Wed., Nov. 14 from 7:30-9:30 in Science Center C & E.
  • A final exam scheduled by the University; the preliminary date is is Tue, Jan. 22.


Last update, 8/30/2001, math21a@fas.harvard.edu