Showing posts tagged M32AH

Compactness and Open Covers

Let \(S \subseteq \mathbf{R}^n\) and suppose \(\mathcal{A} = {A_i | i \in I}\) is a family of open subsets of \(\mathbf{R}^n\) such that \[ S \subseteq \bigcup_{i \in I} A_i. \] We call such a family \(\mathcal{A}\) an open cover of \(S\). We call a subset \(K \subseteq \mathbf{R}^n\) topologically compact if every open cover \(\mathcal{A}\) of \(K\) admits a finite subcover. That is for every open cover \(\mathcal{A}\), there exists \(k \in \mathbf{N}\) such that there exist \(A_1,…

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Math 32A/H Final Review Materials

I have started compiling some review materials for math 32A and 32AH for this quarter. They consist of Review Exercises Solutions I will post solutions to the exercises before the review session on Thursday. As always, let me know in the comments below if anything is unclear or incorrect. Update I have posted the solutions to the practice problems. Also, a student pointed out a typo in statement of the final problem in the review. Spherical coordinates should be given by $$ x = \rho \cos \theta \sin \varphi, \quad y = \rho…

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Math 32AH Final Review

I have written up an overview of much of the material covered in Math 32AH this quarter. Hopefully this review will serve as a good study guide for the course. The bulk of the review is a statement of the definitions of most of the terms defined in class. I didn’t write out any detailed proofs in the review, but left the proofs of many statements as exercises. The exercises therefore tend to be rather abstract and conceptual. Later this week, I will post more review questions (with solutions)…

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32A Midterm 2 Review Materials

I have written up some review materials for the second midterm for Math 32A. They include: Overview Review questions Review solutions I think the questions and solutions cover the majority of the conceptual material likely to appear on the midterm. I’ve taken some of the problems/solutions from previous times I’ve taught this course, so some of the notation may not agree with what you are used to — so be warned. As usual, please let me know in the comments below if anything is unclear or incorrect.…

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Unions and Intersections of Closed Sets

Let \(S \subset \mathbf{R}^n\). We call \(x \in \mathbf{R}^n\) a boundary point of \(S\) if for every \(\varepsilon > 0\) the ball \(B(x, \varepsilon)\) centered at \(x\) of radius \(\varepsilon\) contains at least one point \(y \in S\) and at least one point \(z \notin S\). The set of all boundary points of \(S\) is called the boundary of \(S\) denoted \(\partial S\). We say that \(S\) is closed if \(\partial S \subset S\). Now let \({S_\alpha}\) be a collection of closed sets indexed by…

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