Math 1800: Multivariate calculus #

Functions of more than one variable are ubiquitous: rates of chemical reactions depend on temperature and pressure, crop yields depend on the amount of rainfall and fertilizer used, and most quantities studied in economics depend on dozens of input variables. Multivariate calculus adapts the tools of calculus to this, much richer, setting.

The course starts by developing the language and tools necessary to study multivariate functions, such as graphs, contour diagrams, and vectors. We then develop notions of differentiation and integration as natural analogues of those seen in single-variable calculus. The course ends with the study of new calculus notions unique to the multivariate setting.

Regular coursework is supplemented by laboratory work that introduces students to compelling applications of multivariable calculus and develops intuition for the subject.

Course description: Multivariate calculus in two and three dimensions. Vectors and curves in two and three dimensions; partial and directional derivatives; the gradient; the chain rule in higher dimensions; double and triple integration; polar, cylindrical, and spherical coordinates; line integration; conservative vector fields; and Green’s theorem. An average of four to five hours of class meetings and computer laboratory sessions per week.

Learning goals: Upon completion of this course, a student should be able to:

• graph functions of more than one variable,
• understand vectors, their products, and their geometric meanings,
• differentiate and integrate functions of more than one variable,
• understand the underlying geometrical meaning of these operations, and
• apply notions of multivariate calculus to solve problems in the sciences.

Book and Materials #

For a good portion of the class, will closely follow the text Multivariable Calculus by Hughes-Hallett et al., although some course content will only be available from the lectures and laboratories. We will use the eigth edition of the text. You may find a bookstore purchase link on Polaris. So that everyone can start working on the class right away, the first few sections are available. Make sure to secure your own copy as soon as practical.

Reading and watching mathematics #

There will be times this semester where you will be asked to read a section or watch a short video about a topic related to the course. Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say. Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours. In each module, I will let you know how long you should expect to spend reading and watching the material. Some weeks, this will be a substantial commitment of time even before you start the homework.

Course meetings and office hours #

We will meet in Searles 217 every Monday, Wednesday, and Friday at 10:05am. Whenever possible and appropriate, lectures will be recorded for your enjoyment and to help you review your notes. If it becomes necessary this semester, we will meet over Zoom. The link is available on the menu to your left.

I will hold formal office hours on:

There may be times in the semester where I will have to reschedule office hours due to a conflict. I will let you know by email beforehand. I am available throughout the week for additional meetings. To schedule either individually or as a group, please send me an email with a couple of times that will work. And please don’t hesitate to set these up, I am very happy to see you.

Homework #

Homework problems will be assigned after every class, and will be due once a week each Wednesday at 9am. Each submission will consist of three parts:

• Cover sheet: In addition to your name, I will also ask you to recognize individuals you worked with and sources you used to complete your work. There will also be a space to briefly discuss your group meetings that week.

• Class notes: As you participate in class, I expect you to actively take notes. You will submit them as part of the homework each week. They will be graded generously, but should be complete and legible.

• Homework problems: Each week, we will work on several sections of the book. Problems from each section should be submitted on separate pages. Please leave ample room for grader comments.

Your homework will be graded using GradeScope. To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you edit your homework electronically, make sure you can save or export your work in this format.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the help document for a list of suggestions. Please let me know if this does not work for you; I will come up with an alternative.

Late work: Late work will generally not be accepted as I post solutions immediately after the work is due. However, I will drop your lowest three homework scores.

Collaboration and groups #

Throughout the course of the class, you will be a part of a group. While your work will be written-up individually, I would like all of you to check in with each other and discuss the course material and homework at least once a week outside of class.

Laboratories #

Labs will be (hopefully) fun, address interesting practical and theoretical questions, occur irregularly during the semester, and done collaboratively. You will need to bring a device to class: either a laptop or tablet will do. More details to come.

Exams #

There will be three exams during the semester, tentatively scheduled for:

• Friday, March 3rd,
• Friday, April 21st, and
• a comprehensive exam at 8:30am Monday, May 15th.

There will be an evening question and answer session a day or two before each exam. More details will be announced in class.

Course Assessment #

Your grade will be based on the homework (25%), exams (70%), and class engagement (5%). The class engagement part of your grade is a means for me to recognize your contribution to the education of others in the class. There are many ways to contribute and everyone should find a way that works best for them. Here are some possible ways to do this:

• contribute during lectures: questions, answers, or perspective;
• participate actively in your group;
• and make sure to recognize the contributions of others on the homework cover sheets.

At the end of course, I will ask you to fill out a brief form enunciating your contributions in this area.

Password #

A password will be required to access some features of the course. I will announce it in class, but if you forget it, send me a quick email.

Some axioms #

Federico Ardila enunciated the following axioms in his Todos Cuendan. They form a lens through with I view both teaching and doing mathematics.

• Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
• Everyone can have joyful, meaningful, and empowering mathematical experiences.
• Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
• Every student deserves to be treated with dignity and respect.