Syllabus
Course name
Multivariable Calculus Math 21a,, Harvard College/GSA: 6760, Fall 2013/2014, Exam group 1This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Course head
Oliver Knill, knill@math.harvard.edu, SC 432, Harvard UniversityMeeting time
After a short intro meeting on Wednesday September 4 at 8:309:00 AM in Sci Center B which all students attend, classes are taught in sections on MWF 9,MWF 10,MWF 11, MWF 12, MWF 2PM TuTh 1011:30, TuTh 11:301. Our "teaching in section" approach has been praised for the last decade. Yes, it is expensive and requires coordination but it has proven to build closer relation ship with the instructor, allowing discussing the material in a Socratic manner, also one by one in office hours and have instructors knowing you in person. There are global reviews and a Mathematica workshop which are taught for the entire class. Classes in individual sections start on Monday, September 9. More information about sectioning.Problem sessions
Course assistants will run additional problems sessions. It is highly recommended to use them for discussing the material more.Office hours
Office hours of all the teaching staff will be posted. You are welcome to join any of the office hours.MQC
The Math question center (MQC) is a place, where you can hang out to work on your course work. The MQC takes place SunThu 8:3010:30 PM in SC 309a. The MQC will open on Sunday, September 8, 2013 and stay open until December 5 2013, the first evening of the 2013/2014 reading period.Prerequisite
A solid single variable calculus background is required. The mathematics department provides advising if you are unsure. You can also check with the course head of this course.The course
. Multivariable calculus is a fundamental pillar for many other things: It extends single variable calculus to higher dimensions. You will see that the structures are much richer than in single variable and that the fundamental theorem of calculus generalizes to higher dimensions.It provides vocabulary for understanding fundamental processes and phenomena. Examples are planetary motion, economics, waves, heat, finance, epidemiology, quantum mechanics or optimization.
It teaches important background needed in social sciences, life sciences and economics. But it is rigorous enough that it is also suited for students in core sciences like physics, mathematics or computer science.
It builds tools for describing geometrical objects like curves, surfaces, solids and intuition which is needed in other fields like linear algebra or data analysis. Geometry is currently extremely hot: tomography methods in medicine, computer games, google earth, social network analysis all use geometry.
It relates to culture and history. The quest for answering questions like "where do we come from", "what will future bring us", "how can we optimize quantities" all use calculus. They were the motor to develop it. Euler, the inventor graph theory for example knew geometry and calculus well. The history of calculus contains fascinating stories, starting from Archimedes, 2300 years ago up to the modern times, where new branches of multivariable calculus are developed to understand the structure of nature. It develops methods for solving problems. Examples are optimization problems with and without constraints (which is the bread and butter for exconomics), geometric problems, computations with scalar and vector fields, area and volume computations.
It makes you acquainted with a powerful computer algebra system which allows you to see the mathematics from a different perspective. Such systems are more and more needed for visualization, experimentation and to build laboratories for your own research. No programming experience is required however. We will provide templates and get you started with a workshop.
It prepares you for further study in other fields. Not only in mathematics and its applications, but also in seemingly unrelated fields like game theory, probability theory, discrete mathematics, sociology, or number theory, where similar structures and problems appear, even in a discrete setting. Without geometric intuition and paradigms learned in calculus, it is rather hard to work in those fields.
It improves thinking skills, problem solving skills, visualization skills as well as computing skills. You will see the power of logical thinking and deduction and why mathematics is timeless.
Lectures:
The lecture times are MWF 9, MWF 10, MWF 11, MWF 12, MWF 2 PM (a first this year), TuTh 1011:30, TuTh 11:301. The sections are all coordinated and teach the same material. Learning it in a smaller class helps you to absorb it better and to learn more efficiently. You will section for this course online. The actual lectures start on Monday, September 9. Tuesday/Thursday sections start on Tuesday, September 10.Text
Virtually all calculus books have the same structure. We follow Multivariable Calculus: Concepts and Contexts, 4 book by James Stewart: for example the fourth edition. The edition is not that important and even an other book works. Stewart Multivariable Calculus Edition 4E has the ISBN number ISBN13:9780495560548 or the "fat version" ISBN10: 0495557420 are fine. Also newer or older editions work. A copy also in the Cabot library on reserve.Exams
There are two midterm exams and one final exam. First hourly: Wednesday, October 2, 2013 Emerson 105, 78:30
 Second hourly: Thursday, November 7, 2013 Emerson 105, 78:30
 The final exam takes place December 16, 2013 at 2 PM Schedule.
Grades
First and second hourly 30 % total Homework 25 % Mathematica project 5 % Final 40 %  Final grade 100 % 
Graduate Credit
This course can be taken for graduate credit for almost all graduate schools. We regularly have graduate students taking our multivariable calculus course. If in doubt, check with your school before the semester starts whether you get credit. To fulfill the graduate credit requirements, a minimal 2/3 score must be reached for the final project.Academic integrity
The usual rules outlined in the student handbook apply. As outlined in the Grades section, submitted work consists of: 1. Homework Discussing homework with section leaders, CAs and fellow students is encouraged but submitted work must be written down by each student. When using computer algebra system or online tools for homework the use needs to be acknowledged in the paper. If you receive homework help from anyone other than the course TFs and CAs, please acknowledge that help also at the top of your assignment; your homework score will not be affected by this. We encourage that you work on homework first yourself and without computers to see, where difficulties are and to be prepared for exams. You can use office hours, MQC or problem sessions to overcome obstacles and to deepen the understanding.
 2. Two midterm exams and a final. Exams are all closed book and done without electronic aids or discussion.
 3. The Mathematica project is of creative nature and does not require any programming knowledge. Templates will be provided so that this can be done in modest time. You are encouraged to consult with faculty, course assistants and fellow students to overcome technical issues or suggestions, but you will submit your own work. More about Mathematica in the next section.
Mathematica project
The course traditionally features a Mathematica project, which introduces you to the advanced and industrial strength, computer algebra system. It is an extremely high level programming language also: where objects can be almost anything: a picture, a book, a website, a social network or a movie. The commandA = Import["http://www.fas.harvard.edu/~math21a/syllabus.html"]; n=StringCount[A, "Harvard"]; Speak[A];for example reads in the syllabus of this course, counts the number of words "Harvard" and reads the text to you. This semester, we do something special again. Details and the assignment will be posted later. Mathematica for which Harvard has a site license. At the end of the semester you submit a short project. The actual lab will be posted later in the semester. This software does not lead to any additional expenses and the total time for doing the lab is of the order of a homework problem if you do the minimal requirement.
Calendar
FAS Calendar
FAS 5 year calendar.
 Su Mo Tu We Th Fr Sa Event  1 2 3 4 5 6 7 0 Sep 2: Labor day, Sep 4 Intro 8:30 Hall B 8 9 10 11 12 13 14 1 Sep 9: first MWF class 21a, Sept 10 TTh 15 16 17 18 19 20 21 2 22 23 24 25 26 27 28 3 29 30 1 2 3 4 5 4 Oct 2: first hourly, Emerson 105, 7 PM 6 7 8 9 10 11 12 5 13 14 15 16 17 18 19 6 Oct 14: Columbus day, no classes 20 21 22 23 24 25 26 7 27 28 29 30 31 1 2 8 3 4 5 6 7 8 9 9 Nov 7: second hourly, Emerson 105, 7 PM 10 11 12 13 14 15 16 10 Classes on Veterans day Nov 11! 17 18 19 20 21 22 23 11 24 25 26 27 28 29 30 12 Nov 27 Dec 1 Thanksgiving 1 2 3 4 5 6 7 13 Dec 4, last day of class 8 9 10 11 12 13 14 14 Dec 5Dec 11 Reading period 15 16 17 18 19 20 21 15 Exam period ends Dec 20 
Day to day lecture
We cover chapters 913 in the Stewart book.

Hour Topic Book section 1. Vector geometry 9/9  9/13 1  coordinates and distance 9.1 2  vectors and dot product 9.23 3  cross product and planes 9.4 2. Functions 9/16  9/20 1  lines and planes, distances, 9.5 2  level surfaces and quadrics 9.6 3  curves, velocity, acceleration 10.12 3. Curves 9/23  9/27 1  arc length and curvature 10.34 2  other coordinates 9.7 3  parametric surfaces 10.5 4. Surfaces 9/30  10/4 1  review for first hourly on Oct 2 or 3 2  functions and continuity 11.12 3  partial derivatives/gradient 11.3 5. Partial derivatives 10/7  10/11 1  partial differential equations 11.3 2  linear approximation 11.4 3  chain rule,implicit different. 11.5 6. Gradient 10/14  10/18 1  Columbus day, no classes 2  tangent spaces 11.4 11.6 3  directional derivative 11.6 7. Extrema 10/21  10/25 1  maxima, minima, saddle points 11.7 2  Lagrange multipliers 11.8 3  Global extremal problems 11.8 8. Double Integrals 10/28  11/1 1  double integrals 12.23 2  polar integration 12.4 3  surface area 12.6 9. Triple integrals 11/4  11/8 1  triple integrals 12.7 2  review for second hourly Nov 7 3  spherical integration 12.8 10. Line integral theorem 11/11  11/15 1  vector fields (Veterans day ) 13.1 2  line integrals 13.2 3  line integral theorem 13.3 11. Greens theorem 11/18  11/22 1  Greens theorem 13.4 2  div, curl 13.513.6 2  flux integrals 13.6 12. Stokes theorem 11/24  11/29 1  Stokes theorem 13.7 2 Thanksgiving break (no class) Nov 27 Dec 1 3 Thanksgiving break (no class) 13. Divergence theorem 12/2  12/4 1  Divergence theorem 13.5 2  Overview 13.6 December 4: last day of class 