Calculus IV, AMATH231, in Fall 2020

  • Term: Fall 2020.
  • Course code: AMATH231
  • Prerequisite: MATH 237/247, Calculus III
  • Instructor: Achim Kempf
  • Teaching Assistants: Kaiwen (Kevin) Jiang, Richard Lopp, Dinesh Siewnarine
  • Lecture times and location: 

    Videos will be posted Mondays, Wednesdays, Fridays by the end of the day, see below for the links.
     
  • ​First lecture: Wednesday, 9 September 2020
  • Office hours with Prof: Fridays 1-2pm via Zoom
  • Tutorials with the TAs (Kaiwan, Richard or Dinesh): 
     
  • Instead of email, this course uses the online discussion forum Piazza for use by students, TAs and prof:
     
    • The link is here: PiazzaPlease 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.   
  • Grades: Homework 20%, Midterm 30%, Final 50%. 
  • Midterm exam: 
     
    • Logistics:
       
      • Friday October 23rd
      • Time of day, duration and procedure will be announced here closer to the date.
      • You will need your Watcard.
    • Content: 
       
      • All the material in the textbook up until reading week is examinable. 
      • This means all up to and including Sec. 3.2.1 (i.e., up to the top of page 77 in the notes).   
      • Important: Be prepared that some of the questions may be of this format:
         
        • (a) Give a simple example of ABC
        • (b) For the example you gave in (a), calculate XYZ.
        • The example is expected to be your own example, i.e., it should be at least a little different from the examples in homeworks and lecture notes.   
  • Final exam: Day and logistics will be announced here.
     

Lecture videos, lecture notes and homework

Cumulative Lecture notes: Chapters 1-4 (I will post Ch.5 later. I am still editing Ch.5)

Wed 9 Sep: Lecture 1, Video 1, Notes Ch. 1.1-1.1.2   

Fri 11 Sep: Lecture 2, Video 2, Notes Ch. 1.1.3-1.1.5

Mon 14 Sep: Lecture 3, Video 3, Notes Ch. 1.1.5-1.2.1

Wed 16 Sep: Lecture 4, Video 4, Notes Ch. 1.2.2

Fri 18 Sep: Lecture 5, Video 5Notes Ch. 2.1

Mon 21 Sep: Lecture 6, Video 6, Notes Ch.2.2.1-2.2.3

Wed 23 Sep: Lecture 7, Video 7, Notes Ch.2.2.3-2.3.1

Fri 25 Sep: Lecture 8, Video 8, Notes Ch.2.3.1-2.3.3  

Mon 28 Sep: Lecture 9, Video 9, Notes Ch.2.4.1

Wed 30 Sep: Lecture 10, Video 10, Notes Ch.2.4.2  

Fri 2 Oct: Lecture 11, Video 11, Notes Ch.2.5

Mon 5 Oct: Lecture 12, Video 12, Notes Ch.3.1.1

Wed 7 Oct: Lecture 13, Video 13, Notes Ch.3.1.2-3.1.3

Fri 9 Oct: Lecture 14, Video 14, Notes Ch.3.1.4-3.2.1

Mon 19 Oct: Lecture 15, Video 15, Notes Ch.3.2.2-4.1.1

Wed 21 Oct: Lecture 16, Video 16, Notes Ch.4.1.1-4.1.3

Homework (the solutions are to be submitted via Crowdmark):

  • Homework 1. These are warm-up questions based on Calculus I-III. Not to be handed in. 
  • Homework 2. To be submitted via Crowdmark by Monday September 21 at 9:00pm.
  • Homework 3. To be submitted via Crowdmark by Monday, September 28 at 9:00pm.
  • Homework 4. To be submitted via Crowdmark by Wednesday, October 7 at 9:00pm. 
  • Homework 5 . To be submitted via Crowdmark by Monday, October 19 at 9:00pm.

Examinable material

The lecture notes will be self-contained and only the materials covered in the lecture notes and lectures 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. 

Content

In brief, we complete continuous 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. 

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 - 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 structures. 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.