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%. 
     
  • Final exam: Here is the file of the Final Exam Questions

    • Tuesday, 15 December at 12:00-1:40pm + 40min. This means that the upload deadline for the complete set of solutions is 2:20pm.
    • Logistics will be the same as that of the midterm:
       
    • Logistics:
       
      • At 12:00pm, the Final Exam will be posted here, and also announced via email, Piazza and Crowdmark.
      • The solutions need to be handwritten on paper.
      • Each page needs to be uploaded to Crowdmark twice: once plain, and once as a selfie with your Watcard:  
         
        • (A) Each written page is to be photographed alone (same as for homework).
        • (B) Each written page is also to be photographed as a selfie with you, the page and your Watcard or other official ID such as your driver's license.
        • Examples of photos: Photo of Type A and Photo Type B. Make sure to always upload both!
        • Final deadline for complete solutions: 2:20pm. I assume that you have all mastered the technology for uploading to Crowdmark by now.
      • Important: The Final Exam comes with detailed instructions on the first page. 
         
    • Here is the pdf file of the Final Exam.
       
  • Midterm exam: 
     
    • The Midterm Questions are here.
      You have until 1:40pm to submit the solutions to Crowdmark.
    • Logistics:
       
      • Friday October 23rd, 12:00-1:30pm
      • At 12:00pm, the midterm problem sheet will be posted here, and also announced via email, Piazza and Crowdmark.
      • Upload by 1:30pm to Crowdmark. I estimate that the exam itself takes about 50min. So you have 40min for the upload. 
      • The solutions need to be handwritten on paper.
      • Each page needs to be uploaded to Crowdmark twice, once plain, and once as a selfie with your Watcard:  
         
        • (A) Each written page should be photographed alone (same as for homework).
        • (B) Each written page should also be photographed as a selfie with you, the page and your Watcard.
        • Examples of photos: Photo of Type A and Photo Type B. Make sure to always upload both!
      • Important: The midterm comes with detailed instructions on the first page. 
         
    • Midterm content: 
       

Lecture videos, lecture notes and homework

The complete Lecture Notes are here.

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

Fri 23 Oct: Lecture 17, Video 17, Notes Ch.4.1.3

Mon 26 Oct: Lecture 18, Video 18, Notes Ch.4.2.1

Wed 28 Oct: Lecture 19, Video 19, Notes Ch.4.2.2-4.2.3

Fri 30 Oct: Lecture 20, Video 20, Notes Ch.4.3.1

Mon 2 Nov: Lecture 21, Video 21, Notes Ch.4.3.2-4.4.1

Wed 4 Nov: Lecture 22, Video 22, Notes Ch.4.4.1-4.4.2

Fri 6 Nov: Lecture 23, Video 23, Notes Ch.5.1.1

Mon 9 Nov: Lecture 24, Video 24, Notes Ch.5.1.1-5.1.2

Wed 11 Nov: Lecture 25, Video 25, Notes Ch.5.1.2-5.1.3

Fri 13 Nov: Lecture 26, Video 26, Notes Ch.5.1.3-5.1.4

Mon 16 Nov: Lecture 27, Video 27, Notes Ch.5.1.4-5.2

Wed 18 Nov: Lecture 28, Video 28, Notes Ch.5.2.1-5.2.2

Fri 20 Nov: Lecture 29, Video 29, Notes Ch.5.2.2-5.2.3

Mon 23 Nov: Lecture 30, Video 30, Notes Ch.5.2.3

Wed 25 Nov: Lecture 31, Video 31, Notes Ch.5.2.4-5.3.1

Fri 27 Nov: Lecture 32, Video 32, Notes Ch.5.3.2-5.4.1

Mon 30 Nov: Lecture 33, Video 33, Notes Ch.5.4.2-5.4.3

Wed 2 Dec: Lecture 34, Video 34, Notes Ch.5.4.3-5.4.4

Fri 4 Dec: Lecture 35, Video 35, Notes Ch.5.4.5-5.4.6

Mon 7 Dec: Lecture 36, Video 36Complete Notes

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.
  • Homework 6. To be submitted via Crowdmark by Monday, November 2 at 9:00pm. 
  • Homework 7. To be submitted via Crowdmark by Friday, November 13 at 9:00pm.
  • Homework 8. To be submitted via Crowdmark by Wednesday, November 25 at 9:00pm.
  • Homework 9. To be submitted via Crowdmark by Monday, December 7th at 9:00pm (last day of classes).
  • Homework 10. Exercise only (on Fourier transforms), not to be handed in.

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.