Day 32 in Mat430

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Today:

Announcements
- Today's activity, and new assignment:
  Mon 4/5 Wrap up conformal mapping; begin integration
  
  Read sections 38-40
  Problems: pp. 115, #1, 3, 5, 7 (due Mon., 4/12)
Last time: Steady state temperature and conformal mapping:
- Section 100: Steady State Temperature distributions
- Section 101: An Example
Today: finish example of heat conduction, and begin derivatives
- Section 102: A related problem
  - Derive the form of the isotherms
  - Discuss other "easy" problems, using Appendix 2. What's an easy temperature distribution to solve for?
    - How about a rectangular box with two opposing insulated sides, and constant values on the other sides?
    - How about a rectangular box insulated on all sides? (Really easy! -- where will it equilibrate?)
    - Now what transforms into a finite rectangular box, and what can a finite rectangular box turn into?
- Section 36
  - We're working with parametric curves
  - Many of the usual rules work
  - Some don't, however (e.g. Mean Value Theorem)
- Section 37
  - Anti-derivatives are working for these parametric curves
  - Complex integration allows us to get a "twofer" sometimes. As an example, we've got ${\int_{0}^{\pi}e^{(1+i)x}dx=\int_{0}^{\pi}e^{x}cos(x)dx+i\int_{0}^{\pi}e^{x}sin(x)dx}$
  - An inequality

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