Explorations in analysis, topology, and dynamics : an introduction to abstract mathematics / Alejandro Uribe A., Daniel A. Visscher

By:

Uribe Ahumada, Alejandro [VerfasserIn]

Contributor(s):

Visscher, Daniel Alan, 1984- [VerfasserIn]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Pure and Applied Undergraduate Texts ; 44Publisher: Providence, Rhode Island : American Mathematical Society, [2020]Description: 1 Online-Ressource (1 online resource)ISBN:

1470456850
9781470456856

Subject(s):

Additional physical formats: 1470452707 | 9781470452704 | Erscheint auch als: Explorations in analysis, topology, and dynamics. Druck-Ausgabe Providence, Rhode Island : American Mathematical Society, [2020]DDC classification:

LOC classification:

QA300

Online resources:

Zugang im Netz des KIT

Summary: Cover -- Title Page -- Preface -- Chapter 1. Real Numbers and Sequences -- 1.1. What are the real numbers? -- 1.2. A first look at sequences -- 1.3. A first look at series -- 1.4. Properties of sequences and series -- 1.5. Subsequences and the Bolzano-Weierstrass Theorem -- 1.6. Sequences and convergence in higher dimensions -- Chapter 2. An Introduction to Point-Set Topology and Continuity -- 2.1. The closure of a set -- closed sets and open sets -- 2.2. Compact sets -- 2.3. Continuous functions -- 2.4. Continuous images of compact sets -- 2.5. Continuous images of intervalsSummary: 2.6. Continuous mappings into bR ^{ } -- Chapter 3. Differential Calculus -- 3.1. The derivative -- 3.2. Using derivatives to find maxima and minima -- 3.3. The Mean Value Theorem -- 3.4. Rules for differentiation -- 3.5. Taylor's Theorem -- 3.6. Power series expansions of a few common functions -- Chapter 4. Integral Calculus -- 4.1. The Riemann integral -- 4.2. The First Fundamental Theorem of Calculus (a telescoping sum) -- 4.3. Change of variables for integration -- 4.4. Averages and the Mean Value Theorem for Integrals -- 4.5. Accumulation and the Second Fundamental Theorem of CalculusSummary: 4.6. Methods for finding antiderivatives -- Chapter 5. Discrete Dynamical Systems -- 5.1. Iterating functions and types of orbits -- 5.2. The logistic map, modeling, and bifurcations -- 5.3. The doubling map and chaos -- 5.4. The tent map and fractals -- 5.5. The rotation map and Benford's Law -- 5.6. The billiard map and phase space -- Chapter 6. Iterating Algorithms and Representations of Real Numbers -- 6.1. Iterating algorithms -- 6.2. Decimals and binaries -- 6.3. The Gauss map and continued fractions -- 6.4. The Euclidean algorithm and inscribing squares in rectanglesSummary: Appendix A. Definitions, Proofs, and Mathematical Language -- A.1. Writing mathematics -- A.2. Writing definitions -- A.3. Writing proofs -- Appendix B. Sets and Functions between Sets -- B.1. The language of set theory -- B.2. Functions between sets -- Appendix C. Graphs -- Appendix D. Hints to Selected Problems -- Index -- Titles in Series -- Back CoverSummary: This book is an introduction to the theory of calculus in the style of inquiry-based learning. The text guides students through the process of making mathematical ideas rigorous, from investigations and problems to definitions and proofs. The format allows for various levels of rigor as negotiated between instructor and students, and the text can be of use in a theoretically oriented calculus course or an analysis course that develops rigor gradually. Material on topology (e.g., of higher dimensional Euclidean spaces) and discrete dynamical systems can be used as excursions within a study of aPPN: PPN: 1755153562Package identifier: Produktsigel: BSZ-4-NLEBK-KAUB | ZDB-4-NLEBK

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