Syllabus
Course name
Multivariable Calculus Math 21a,, Harvard College/GSA: 6760, Fall 2015/2015, Exam group 1This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning (previously called 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 2 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, TuTh 1011:30, TuTh 11:301. Our "teaching in section" approach has been praised for the last decade. It is more expensive and requires coordination but it has proven to build closer relationships 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 course. Classes in individual sections start on Monday, September 8, 2015. More information about sectioning.Problem sessions
Course assistants will run additional problems sessions. It is highly recommended to use this time for discussing the material.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 7, 2015 and stay open until December 4 2015, the first evening of the 2015/2015 reading period.Prerequisite
A solid single variable calculus background is required. The mathematics department provides advising resources 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 problem solving methods. 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, 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 8. Tuesday/Thursday sections start on Tuesday, September 9.Text
Virtually all multivariable calculus books have the same structure. An example is Multivariable Calculus: Concepts and Contexts, 4 book by James Stewart fourth edition. The edition is not important and even an other book works. Stewart Multivariable Calculus Edition 4E has the ISBN number ISBN13:9780495560548. The version which includes single variable is ISBN10: 0495557420. Also newer or older editions work. The Cabot library has a copy on reserve.Exams
There are two midterm exams and one final exam. First hourly: Tuesday, September 29, 2015 at 8 PM
 Second hourly: Tuesday, November 2, 2015 at 7 PM
 The final exam takes place in December 2015 and is organized by the exams office. The final exam date will be announced by the registrar.
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. Details and the assignment will be posted later. Mathematica for which Harvard has a site license (currently, the latest edition is Mathematica 10). 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 5 year calendar.
 Su Mo Tu We Th Fr Sa Event  30 31 1 2 3 4 5 0 Sep 2: Intro 8:30 Hall B, Sep 7 Labor day 6 7 8 9 10 11 12 1 Sep 9: first MWF class 21a, Sept 10 TTh 13 14 15 16 17 18 19 2 20 21 22 23 24 25 26 3 27 28 29 30 1 2 3 4 Sep 29 first hourly, Hall B 4 5 6 7 8 9 10 5 11 12 13 14 15 16 17 6 Oct 12: Columbus day, no classes 18 19 20 21 22 23 24 7 25 26 27 28 29 30 31 8 1 2 3 4 5 6 7 9 Nov 3: second hourly, Hall B 8 9 10 11 12 13 14 10 Nov 11, Veterans day, Classes 15 16 17 18 19 20 21 11 22 23 24 25 26 27 28 12 Nov 25 Nov 29 Thanksgiving 29 30 1 2 3 4 5 13 Dec 2, last day of class 6 7 8 9 10 11 12 14 Dec 3Dec 9 Reading period 13 14 15 16 17 18 19 15 Exam period ends Monday Dec 21 
Day to day lecture

Hour Topic Book section 1. Vector geometry 9/9  9/12 Labor day 1  coordinates and distance 9.1 2  vectors and dot product 9.23 2. Functions 9/14  9/18 1  cross product, lines planes 9.49.5  distances 2  level surfaces and quadrics 9.6 3  curves, velocity, acceleration 10.12 3. Curves 9/21  9/26 1  arc length and curvature 10.34 2  other coordinates 9.7 3  parametric surfaces 10.5 4. Surfaces 9/28  10/2 1  review for first hourly on Tuesday 2  functions and continuity 11.12 3  partial derivatives/gradient 11.3 5. Partial derivatives 10/5  10/9 1  partial differential equations 11.3 2  linear approximation 11.4 3  chain rule,implicit different. 11.5 6. Gradient 10/12  10/16  Columbus day, no classes 1  tangent spaces 11.4 11.6 2  directional derivative 11.6 7. Extrema 10/19  10/23 1  maxima, minima, saddle points 11.7 2  Lagrange multipliers 11.8 3  Global extremal problems 11.8 8. Double Integrals 10/26  10/30 1  double integrals 12.23 2  polar integration 12.4 3  surface area 12.6 9. Triple integrals 11/2  11/6 1  review for second hourly Nov 4 2  triple integrals 12.7 3  spherical integration 12.8 10. Line integral theorem 11/9  11/13 1  vector fields (Veterans day ) 13.1 2  line integrals 13.2 3  line integral theorem 13.3 11. Green and Stokes theorem 11/17  11/21 1  Greens theorem 13.4 2  div, curl, flux 13.513.6 3  Stokes theorem 13.6 12. Divergence theorem 11/23  11/28 1  Divergence theorem 13.7 2 Thanksgiving break (no class) Nov 25 Nov 29 3 Thanksgiving break (no class) 13. Integral theorems Overview 12/2  12/4 1  GreenStokesGauss 13.78 2  Overview 13.58 December 3: last day of class 