# Syllabus

#### Course name

Harvard Summer School course Maths 21a, Multivariable Calculus, CRN 30189, Summer 2018This course is a standard multivariable course. It extends single variable calculus to higher dimensions. It provides vocabulary and background for understanding fundamental processes and phenomena in economics, life sciences, finance or chemistry but it is rigorous enough that it is suited for students in physics, mathematics or computer science. It builds tools for describing geometrical objects like curves, surfaces, and solids. It forms intuition which is needed in other fields like computer vision or data visualization. It develops methods for solving problems. We practice this with optimization problems both with and without constraints, geometric problems, computations with vector fields with with area and volume computations. Math 21a acquaints you also with a powerful computer algebra system which allows you to explore without any programming experience to see the mathematics from a different perspective. It prepares you for further study in other fields. Not only in mathematics and its applications, but also in seemingly unrelated fields like statistics, game theory, probability theory, discrete mathematics or number theory, where similar structures and problems appear. The main goal of the course is the fundamental theorem of calculus in higher dimensions, where all the concepts previously built up fit together. While mastering this milestone in mathematics, you harden your raw thinking abilities and see the power of mathematical abstraction. Other disciplines like economics, physics or life sciences often require this course because it is useful to be able to think mathematically and geometrically. Like any calculus course, multivariable calculus improves thinking skills, problem solving skills, visualization skills as well as computation skills.

#### Instructor

Oliver Knill, knill@math.harvard.edu, SC 432, Harvard University#### Course Assistants

Nicolas Robles nirobles@illinois.edu Andy Li andyli@college.harvard.edu

#### Lecture time

Tuesday/Thursday 8:30 - 11:30 in Harvard Hall 104. A weekly problem section takes place on Wednesday and Thursday afternoons from 2-3, in SC 309a. We always start at 8:30 AM sharp for the lectures (no 7 minutes delay as custom during the semester at Harvard). We will make a short break after 90 minutes. Since July 4 hits a Tuesday, we will have our review lecture on the Monday July 3, from 7-9 PM in Lecture Hall C. This is also when the homework for the second wee is due!#### Problem sessions

Additional to the lectures there is a problem session on Thursday Wednesday or Thursday 2-3 PM. These sections are not mandatory but they are highly recommended.#### Prerequisite

A solid background in single variable calculus is required. If you want to look some old things up, here is some material from a single variable course. A 15 minute refresher. And here is a 20x20 sec Pecha-Kucha presentation of calculus in a different way.#### The course

Here are the main points again, but in bullet points:**It extends single variable calculus**to higher dimensions. You will see that the structures are much richer and that the fundamental theorem of calculus comes in two flavors in dimension 2 and has three incarnations in dimension 3.

**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.

**It develops methods**for solving problems. Examples are optimization problems with and without constraints, 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. No programming experience is required however.

**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 or number theory, where similar structures and problems appear.

**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. The insight you gain this summer will remain true and relevant forever because mathematical truth is eternal.

#### The workload

For this standard multivariable calculus course, the total lecture time is roughly the same as for our standard Math21a course in the Harvard semester. We cover the same material but do this in a slightly more streamlined way. This course is a "tough" course. It requires about 20 hours of work per week in average from a typical student. This estimate includes lectures, recitation, reading, homework and exam preparation. Calculus is not only an important subject, it is a prototype theory which shows**how to think spatially**. This is pivotal for example for a surgeon or when illustrating data. Because multivariable calculus has as a well defined goal, we climb "the fundamental theorem of calculus in higher dimensions", it a benchmark theory for which there is hardly any short cut.

If you take two heavy classes (like a 21a/21b combination in the summer school) simultaneously, this leads to a 40 hour work week. It is possible, but demanding. About a dozen students do it every year. Taking two courses only leaves time for enjoying the summer in Cambridge if the work time is well managed. If you have a half time job, a heavy sports or music schedule or if you want to reserve time for a personal project, I recommend you take this course only without a second course.

#### Lectures:

Lectures take place every Tuesday and Thursday at 8:30-11:30, in Harvard Hall 104. The course time is condensed to 6 weeks with 6 hours of lecture every week. The 3 hour lectures can be tough (no for me giving the lectures but for you to listen). We try to make it entertaining although. You can bring a snack since we always will have a short break after 90 minutes. I divide the material of the course into 24 units. Two units of 90 minutes are taught every Tuesday and Thursday. Classes which also feature the exam are a bit condensed.#### Office hours

Oliver: Monday 15:00-17:00, in SC 432.#### Text

All course material and homework will be handed out in class. In principle, you do not need a textbook. Reading a textbook can give a "second opinion" on the material. A widely used textbook is "Multivariable Calculus: Concepts and Contexts" by James Stewart. Any multivariable text works; they are all very similar to each other. Especially, any old edition does fine. We cover the material which can be found in chapters 9-12 of Stewart or chapters 10-14 in Varberg Purcell Rigdon or chapters 10-15 in Smith Minton.#### Homework

Homework problems are included on the lecture notes. Each lecture unit has 5 problems so that you do 20 problems each week. Homework will be returned weekly. Homework is due on Tuesdays except for the last week, where the homework is due Tuesday and Thursday. It is strongly recommended that you do the first 10 problems after the first day. This will also make it more easier to make sense the second day each week.#### Exams

There are two midterm exams and one final exam. The midterms are administered during class time in the usual lecture hall Harvard Hall 104. The final exam will take place during the examination period, again in the usual lecture hall. Due to the summer school schedule you will naturally not have much time to prepare for the exams. It will therefore be important to keep always up with the material and ask if something is not clear.#### Grades

First and second hourly 40 % total Homework 25 % Project 5 % Final 30 %Active class participation and attendance can boost your final grade by up to 5%.

#### Graduate Credit

This course can be taken for graduate credit. The course work is the same. To fulfill the graduate credit requirements, a minimal 2/3 score must be reached for the final Mathematica project.#### Mathematica project

The use of computers and other electronic aids can not permitted during exams. A Mathematica project will teach you the basics of a computer algebra system Mathematica 10 for which Harvard has a site license. This software does**not**lead to any additional expenses and the total time for doing the lab is 3-4 hours. The project will be handed in at the beginning of the August 2 lecture.

#### Calendar

+----+ +----+ Su Mo | Tu | We | Th | Fr Sa |Week Event ------+----+----+----+--------|------------------------- 24 25 | 26 | 27 | 28 | 29 30 |1 June 26 Start of lectures 1 2 | 3 | 4 | 5 | 6 7 |2 Jul 4 Independence Day 8 9 | 10 | 11 | 12 | 13 14 |3 Jul 12, First hourly 15 16 | 17 | 18 | 19 | 20 21 |4 22 23 | 24 | 25 | 26 | 27 28 |5 Jul 26, Second hourly 29 30 | 31 | 1 | 2 | 3 4 |6 Project due Aug 2, 5 6 | 7 | 8 | 9 | 10 11 |7 Thursday, August 9, Exam +----+ +----+ +--------------------------

#### Day to day lecture plan 2018

1. Week: Geometry and SpaceLect 1-2 6/26 Space, Vectors, Dot Product Lect 3-4 6/28 Cross product, Lines/Planes, Distances2. Week: Surfaces and CurvesLect 5-6 7/3 Implicit and Parametric Surfaces Lect 7-8 7/5 Curves, Chain Rule, Arc Length3. Week: Gradient and LinearizationLect 9-10 7/10 Partial Derivatives, Review Lect 11-12 7/12 First hourly. Linearization Tangents4. Week: Extrema and Double IntegralsLect 13-14 7/17 Extrema and Lagrange Lect 15-16 7/19 Double integrals and Surface area5. Week: Multiple Integrals and Line IntegralsLect 17-18 7/24 Triple integrals also in Spherical coordinates Lect 19-20 7/26 Second hourly. Line integrals6. Week: Vector Fields and Integral TheoremsLect 21-22 7/31 Curl, Greens theorem, Flux Lect 23-24 8/2 Stokes and Divergence theorem7. Week: Review and Final examReview 8/7 Review and Mathematica project due Final 8/9 Final three hour exam