assigned today:

Due today:

Due today:

- standard-complex functions:
I can manipulate complex functions.

- Show how a complex function can produce a vector field.
- Derive the Cauchy-Riemann equations.
- Problem 13.4.25

- Quiz on problem 13.4.25. Gave \(u(x,y)=x^2-y^2\) and asked for v.
- we talked about how plotting the equilines for v is easier than for u.

- They voted for what of the 6 complex problems are the hardest (really they ranked them). The winner seemed to be the \(\sqrt[n]{a+i b}\) problem.
- We considered the Laplacian equation of harmonic functions and talked about the ramifications of them.
- I don't think this went well. I need to structure a better exercise for this. I thought about proving the mean value theorem for harmonic functions but didn't really think there was time.

- I asked them to prove that \(\sqrt[3]{a+i b}\) has at most one real root.
- Some tackled it geometrically, but not all.
- we talked about how 1 has complex cube roots and how weird that was
- We also talked about how the trick (keep adding \(2\pi\)) has an end point. I asked if every believed that as soon as \(6\pi\) gives you a repeat you can be convinced that there's no reason to go any further.

- I asked them to talk about what further resources they thought they might need.
- I found that it helped them focus by trying to figure out problems with tough changes I might make in the quiz.
- Ela asked about how to deal with it if I gave a map and asked for the complex function that produces it.
- I gave them two examples on the board to think about that: constant vectors up at 45 degrees and a upward field that got stronger as you get higher.

No new resources requested.