\begin{equation*} \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\bar}{\overline} \renewcommand{\Lin}{\mathrm{Lin}} \renewcommand{\dd}{\mathrm{d}} \end{equation*}

Instructions

Either

  • Record and submit a video of up to 5 minutes in which you walk the viewer through the solution of the problem given.
  • Pair up with someone else in class, record and submit a video of up to 8 minutes in which you walk the viewer through the solution of the problem given.

    In case you are doing with a partner (encouraged) each of you must roughly speak for half of the time.

General ideas

  • You will have to use most theorems about curl and conservative vector fields. Say the facts that you use. Display the computations that justify your claims (prepare them beforehand). Do not do intermediate steps.
  • Imagine that you are talking to someone that has read the book, but does not know how to actually carry out the solution of the problem.
  • Do NOT read out the mathematical formulas. Make sure they are visible and reference them as "this domain"/ "domain \(\Omega_1\)", "this function" / "function \(f_1(x,y,z)\)"…
  • Clearly state what facts you are using.
  • Do not carry out computations live, but have the main steps on screen.
  • Use visual aids (geogebra/hand-drawn)

Best method to record videos

  • Join your personal Zoom meeting, share screen (your video feed will be in the corner). Start recording. Pause recording when you finish a part of the proof.
  • If doing with a partner, this works really well. Pause the recording half-way, make the other person share the screen, resume recording.
  • Other methods are accepted.

Possible ideas/tips

  • Have a slideshow of your computations Google slides / Powerpoint / Keynote / Libreoffice Impress
  • Include excerpts (cutouts/screenshots) from the book of facts your are using.
  • Start this video by saying: "In this video we study … and show that …" i.e. you can state your claim at the beginning.
  • If you want to post-process the video you can use open-source video editing programs like: Openshot.
  • Prepare additional material like a 3D or 2D graph to aid in communicating your ideas.

Submission

Please submit under "File Drop" in Collab. If you submit earlier than the deadline an attempt will be made to assess your submission earlier and provide feedback.

You will get full marks if you solve the problem and provide an explanation for the solution.

Minor computational mistakes will not negatively impact your result as long as it is clear you understand the methods involved in solving the problem.

You will not be allowed to resubmit this assignment.

Objective

This assignments are in preparation for the second midterm that is going to be an oral exam.

1 Problem

Explain why the field

\begin{equation*} \vec{F}(x,y) = \begin{pmatrix} \frac{y^2}{(x^2+y^2)^{3/2}} \\ \frac{-xy}{(x^2+y^2)^{3/2}} \end{pmatrix} \end{equation*}

on \(\R^2\setminus\{(0,0)\}\) is conservative even though the domain is NOT simply connected.

Find \(f\colon\R^2\setminus\{(0,0)\}\to \R\) such that \(\nabla f = \vec{F}\).

Simply displaying \(f\) and checking \(\nabla f = \vec{F}\) is NOT ENOUGH for this problem. You must explain how you managed construct it using path integrals.

Hints:

  • Show that the curl on the domain is \(0\) everywhere.
  • Compute the circuitation around a circular loop around the origin at radius 1.
  • Show that \(f\) must be constant in the radial direction.
  • Draw pictures.