Fundamentals Of Mathematics (9th Edition)

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Textbook: Lay-Lay-McDonald, Linear Algebraand its Applications (5th and 6th editions) and Nagle-Saff-Snider, Fundamentals of Differential Equations and Boundary Value Problems (9th edition). A specially priced UC Berkeley paperback version (2nd edition) is available containing the chapters of both books needed for the course.

Students will break into small groups to tackle these questions (and any others that might come up) for particular components of this content, and towards the end of class will give a "mini-presentation" to showcase some of those choices. Presentation 1: Conjugation and ModulusPresenter: EttaText sections: Chapter 1, Section 3 (of edition 6). In the seventh edition, this is equivalent to the start of section 4 (up to displayed equation (3)) as well as section 5.Content: Whereas the previous two sections dealt with the algebraic properties of complex numbers, in this section we get our first glimpse into its geometric side. The section introduces the notion of complex conjugation, and shows its connection to the modulus (aka, absolute value, or --- even more intuitively --- "size") of a complex number.Comments: As with the previous section, the layout of the chapter is heavy on exposition, and very light on signposting. The mathematical content of this section is likely not what will be challenging when crafting this presentation. Instead, you'll want to make sure you have a sense for the story you're trying to tell, and you'll also want to think carefully about how you package that information so that it is legible and structured. What theorems do you intend to give complete proofs for, and which ones will you glide over? What will you name theorems and definitions? What examples will you create to illustrate new concepts? Presentation 2: Triangle inequalityPresenter: AvaText sections: Chapter 1, Section 3 (of edition 6). In the seventh edition, this is equivalent to the end of section 4 (all content that follows displayed equation (3) on page 10). In the ninth edition, this is equivalent to section 5 (though in this edition, you don't need to dive deeply into Example 3 which explains why reciprocals of polynomials have a certain bounding property).Content: The triangle inequality --- an analysis student's best friend in Math 302! --- makes it's appearance for the complex numbers.Comments: This section continues to put a lot of time into exposition, but not much into signposting. You should feel free to reshuffle and organize as you see fit to make the content more legible. The mathematical content of this section feels a bit more "analysis-y" than what we've seen up to this point, and you should feel free to lean on students' familiarity with the triangle inequality from Math 302. Presentation 3: Polar coordinate representationsPresenter: MilesText sections: Chapter 1, Section 5 (of edition 6). In the seventh edition, this is equivalent to section 6 of chapter 1. In the ninth edition, this is equivalent to section 7 of chapter 1.Content: Continuing with the theme that elements of $\mathbb{C}$ can be thought of as elements of $\mathbb{R}^2$ (but endowed with some additional algebraic struture), this section highlights the way we can interpret complex numbers in terms of a polar coordinate representation.Comments: One of the most powerful facts about elemnts of $\mathbb{C}$ is that they have a dual nature: they can be thought of in a way that is algebraic, but also in a way that is geometric. That geometric understanding benefits greatly from having a representation of elements of $\mathbb{C}$ that keeps track of "angle" information and "magnitude" information. Part of your goal in this section is to highlight these geometric qualities and make them feel natural and useful. Presentation 4: (Multiplicative) arithmetic in polar coordinatesPresenter: EmilyText sections: Chapter 1, Section 6 (of edition 6). In the seventh edition, this is equivalent to section 7 of chapter 1. In the ninth edition, this is equivalent to sections 8 and 9 of chapter 1. Content: In the previous section we worked to give a polar coordinate representation for complex numbers. In this section, we see how the characteristics of the polar coordinate representation of two complex numbers interrelate to produce the corresponding geometric information for their product and quotient. This is used to give a new explanation for a classical trigonometric identity known as de Moivre's formula.Comments: As before, a big part of the lift here is going to be packaging this information so it is comprehensible and "natural." You will of course need to get the mathematics correct, but don't let narrative fall to the wayside! Presentation 5: Solving equations of the form $z^n=b$ for $b \in \mathbb{C}$Presenter: FatimaText sections: Chapter 1, Section 7 (of edition 6). In the seventh edition, this is equivalent to sections 8 and 9 of chapter 1. In the ninth edition, this is equivalent to sections 10 and 11 of chapter 1.Content: The structure of solutions sets of polynomials of the form $z^n=b$ is determined, and a handful of examples are computed.Comments: This is the first section where we need to dig into some ideas that might seem a bit new and unexpected. There's both theory building as well as some interesting examples. As you structure your presentation, you'll want to think carefully about both. On the theory building side, ask yourself how you can adequately package and signpost the content that is being developed. (What result(s) are you working to prove? Work to state them clearly, and consider changing the narrative structure so that the proof of the result follows the statement of the result.). When you move towards illustrative example(s), spend time thinking about what would be most illuminating. Should you stick with an example from the book? Is it worth creating a different example that showcases something the book's presentation misses?Presentation 6: Regions of $\mathbb{C}$Presenter: CherithText sections: Chapter 1, Section 8 (of edition 6). In the seventh edition, this is equivalent to section 10 of chapter 1. In the ninth edition, his is equivalent to section 12 of chapter 1. Content: The basic topology of $\mathbb{C}$ is discussed.Comments: This section tells us what many of the familiar topological constructions from Math 302 look like in the setting of $\mathbb{C}$. You'll want to be faithful to the definitions that the book presents, but you might try to connect these definitions back to the definitions that folks saw when they took 302. (Though remember: maybe not everyone learned 302 the same way!)Unit 2: Differential calculus for functions of complex numbersNow that we know the basics of what it means to be a complex number and some of the associated properties, we can dive into functions that are related to the complex numbers. Because complex analysis is, after all, an analysis course, we will naturally be interested in questions like: what does it mean to talk about the limit of a complex function? when is a complex function continuous? when is a complex function differentiable? These questions should sound familiar from Math 302, and certainly some of the answers we give in this class will be analogous to what happens in the real case; however, with the Cauchy Riemann equations we will start to see that differentiability in the context of $\mathbb{C}$ is a little bit special. 781b155fdc