# Mathematical and Computational Methods in Physics and Engineering (fall 2016)

## Daily outline for Mon, Sep 19, 2016 9:10 AM (1 year ago)

### Review

assigned today:

Due today:
• standard-complex functions:

I can manipulate complex functions.

1. Show how a complex function can produce a vector field.
2. Derive the Cauchy-Riemann equations.
3. 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.

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