# Open Math Notes

## Resources and inspiration for math instruction and learning

Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes:

• Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision.
• Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal
• Refereed publications at the AMS

Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

## Open Math Notes Advisory Board:

• Karen Vogtmann, Chair | University of Warwick
• Tom Halverson | Macalester College
• Andrew Hwang | College of the Holy Cross
• Robert Lazarsfeld | Stony Brook University
• Mary Pugh | University of Toronto

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Showing 1 - 5 of 221 result(s)
 First Order Partial Differential Equations: a simple approach for beginners: First Order Partial Differential Equations Usually a course on partial differential equations (PDEs) starts with the theory of first order PDEs, which turns out to be quite time consuming for a teacher and difficult for students due to dependence of the proofs on geometry of Monge curves. In this article we present a simpler theory of first order PDEs using only the characteristic curves in the space of independent variables. In addition we discuss existence and uniqueness first with examples and then prove rigorously It has new ideas. Phoolan Prasad · Indian Institute of Science · Date posted: January 9, 2023 Send feedback to the author(s) Applied Differential Equations Lecture Notes: Lecture slides for a course taught from "Fundamentals of Differential Equations" These are a series of lecture slides from a course I taught at the University of Alabama in the Fall of 2022. The course text was "Fundamentals of Differential Equations" 9e, by Kent Nagle, Edward Saff & Richard Snider. These slides were used as a lecture supplement with additional explantation, examples and problems.The following sections were covered:1.1-1.4, 2.2, 2.3, 2.6, 3.2, 3.3, 4.1-4.6, 4.9, 5.2, 7.2-7.6. Overleaf link (for LaTeX): https://www.overleaf.com/read/gbnbzrbbyyqz Brandon Sweeting · The University of Alabama · Date posted: January 9, 2023 · Date revised: January 11, 2023 Send feedback to the author(s) Lectures on Modular Forms and Hecke Operators Much enhanced notes by the second author of a course given by the first author in 1996 Kenneth Ribet · University of California, Berkeley · William Stein · CoCalc · Date posted: November 16, 2022 Send feedback to the author(s) Numerical Mathematics II: Numerical Solution of Ordinary Differential Equations We study explicit and implicit methods for initial value problems.For general Runge-Kutta methods, Butcher's theory of rooted treesis presented in detail, proving Butcher's theorem on conditions for high orders of consistency and convergence. Further topics includecollocation methods, stability theory, stiff equations, asymptotic error expansions, extrapolation methods, and stepsize control. Peter Philip · LMU Munich · Date posted: October 31, 2022 Send feedback to the author(s) Vector Fields and Differential Forms Vector fields and differential forms have very different properties. However a given volume element allows vector fields to be converted into twisted differential (n-1)-forms via contraction. Also a metric tensor allows vector fields to be converted into 1-forms. The ultimate theory works with an arbitrary metric tensor in n dimensions. The exposition includes insightful pictures along with extensive computations. William Faris · University of Arizona · Date posted: October 10, 2022 Send feedback to the author(s)

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Showing 1 - 5 of 221 result(s)