Calculus IV, AMATH231, in Winter 2022

  • Term: Winter 2022.
  • Course code: AMATH231
  • Prerequisite: MATH 237/247, Calculus III
  • Instructor: Achim Kempf
  • Online discussion hours with the Prof: every Tuesday, 12:00-1:00pm 
    The Zoom link is here. Attendance is not mandatory but recommended.
  • Teaching Assistant: Mackenzie Cameron (m28camer@)
  • Online discussion hours with the TA: every Friday, 12:00-1:00pm (starting Fri 7 Jan.)
    The Zoom link is here. Attendance is not mandatory but recommended.
     
  • ​First lecture: Wednesday, 5 January 2022
  • Reading week: no classes Feb.19-27
  • ​Last lecture: Monday, 4 April 2022 (or Tuesday 5 April if we are in the in-person teaching schedule by then)
     
  • We will use the online discussion forum Piazza for use by students, TA and Prof:
     
    • The link is here: Piazza.  Please sign up, it is free.
    • Using Piazza is better than emailing the TAs or the instructor because by using Piazza, all can benefit from the discussion. If you wish, you can post anonymously. 
    • Particularly good answers from students to student questions may receive bonus points.
    • Of course, do not discuss the solutions to a homework exercise that is still due.   
  • One-on-one office hours with prof: by arrangement
     
  • Grades: Homework 30%, Midterm 30%, Final 40%. 
     
  • Assignments:
     
    • There will be 10 assignments.
    • Of the 10, only Assignments 2-9 need to be handed in.
    • Submission of the assignment solutions is via Crowdmark.
    • The Assignment sheets and their due dates are in the detailed schedule below. 
       
    • Special advice: 
       
      • Do solve the assignments on your own. I know how tempting it can be to copy because I remember from when I was a student! But of course, copying solutions to assignments would not be fair. And those who do copy won't be prepared for the midterm and the final. 
  • Midterm exam (10 Feb 2022) and Final exam (8 Apr 2022, 12:30-2:30pm) are online: 
     
    • Open book.
    • You will need your Watcard.
    • You will need the ability to quickly take and upload pictures. For example, if you plan to take the pictures with your phone and then upload them with your laptop, exercise this before the exam so that the technology won't make you waste time in the exam. 
    • You absolutely will need a reliable internet connection for zooming during the exam and for the upload afterwards. Make sure of this beforehand. 
    • To keep things as fair as possible for everyone, the midterm and final feature a type of questions for which copying doesn't work well:
    • In the Midterm and Final, there will be questions that ask you for an example of something and then asks you to calculate something for your example. Concretely, a question could be something like this: "Give an example of a differentiable map that describes a nontrivial 2-dimensional curved surface in R3. Choose a point on the surface and calculate the tangent plane to it. Choose your map, surface and point sufficiently generic so that it is highly unlikely that anyone else has the same example." There will be points for the quality of the example and for the correctness of the calculation.    
    • More details announced in the schedule below. 
       

Lectures and Assignments Schedule:
 

Here are the key points regarding the live lectures of our course:

* You do not need to attend in person if you are not comfortable with being in a room with lots of people. 

* Definitely do not come to the lectures if you have any symptoms or if you test positive. Instead, stay home and take care of your health so that you don't get something like Long Covid. I know students who got Long Covid and they say it's bad. 

* Not attending class will not put you at a disadvantage: I will keep the recordings and lecture notes on our home page. The homework, midterm and final will only cover material that is covered in the lecture notes. When I mention any information about homework, midterm and final in a lecture, I will always also post it on our course home page. 

* I will set up a laptop in the front row to live stream the lectures via Zoom. 

* The schedule for the live lectures is different from that of the recorded lectures. We will have only two live lectures per week but they cover exactly the same material as the three recorded lectures per week. The live lectures are therefore correspondingly a bit longer. Their schedule is:

Tuesdays and Thursdays 10-11:15am in Building E2, Lecture Hall 1732  

The first live lecture was on Tuesday, February 8, 2022. 

* The discussion hours and one-on-one office hours will stay online.  

* If you come to class, always wear a proper mask (at least a surgical mask or better a N95, KN95..) and do make sure that it actually fits. 

* If or when the university decides to drop live lectures and go back to fully online, we will return for our course to how we did things so far, without the live lectures. There is, for example, the possibility that the new Omicron variant BA.2 will create a new infection peak in the next few weeks. 

On the bright side, maybe the Omicron variants will finally end the pandemic. Let's hope so! 

Here is the schedule for viewing the recorded lectures, for the life lectures, for submitting the assignments, and for the midterm:   
  


 




Also: Tuesdays, 12:00-1:00pm: Online discussion hours with the Prof. 

  • The Zoom link is here.
  • Attendance is not mandatory but recommended.  
  • What do we do in the discussion hours? 
     
    • Prof answers your questions about the material and beyond, including clarifications about the lectures or the lecture notes, also about the logistics of the course and the exams etc. We will find out if this or Piazza works best. For now, we'll do both. 
    • Questions about assignments: please raise those with our TA MacKenzie since she marks them.     
       
    • If you have issues to discuss with the prof or TA that you don't want to discuss in public, ask for a one-on-one zoom. 



 

Course Content:
 

In brief, we complete calculus. At the end, we show that, even though it is counter intuitive, continuous structures can be entirely equivalent to discrete structures. Real-life applications are ubiquitous. For example, information can be represented in continuous form (e.g. music) and in discrete form (bits in a file). The Shannon sampling theorem explains when and how both representations can be entirely equivalent (no approximations needed).   

Concretely, we first complete calculus by extending integration and differentiation techniques to curved structures such as curved paths and surfaces. The main concepts and results here are vector fields, line and surface integrals and the three famous theorems in this area: Green's theorem, Gauss' Divergence theorem and Stokes' theorem. Applications, e.g., to physics and engineering are emphasized throughout. The second part of the course introduces Fourier analysis, that is, the remarkable fact that a huge classes of functions, in fact essentially all those functions that occur in engineering and physics, are linear combinations of sine and cosine functions. This, in turn, leads to the Shannon sampling theorem which shows that continuous functions can be equivalent to discrete data. This fact is at the heart of information theory. For example, every phone uses Shannon's theorem to transform back and forth between real-life continuous music or speech or video signals and discrete data in a file.

The lecture notes are self-contained and only the material covered in the lecture notes will be examined.

But of course, you are encouraged to read other textbooks and sources as well. Sometimes it is helpful see things from different perspectives. For example, the standard calculus textbook by J. Stewart has a review of vector calculus with useful exercises. For a more in-depth treatment, have a look, for example, into the textbook Vector Calculus by M. Lovric (Wiley, 2007). There are tons of alternatives.