Open Math Notes

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

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We study explicit and implicit methods for initial value problems.
For general Runge-Kutta methods, Butcher's theory of rooted trees
is presented in detail, proving Butcher's theorem on conditions for high orders of consistency and convergence. Further topics include
collocation methods, stability theory, stiff equations, asymptotic error expansions, extrapolation methods, and stepsize control.

Peter Philip · LMU Munich · Date posted: October 31, 2022

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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

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Functional analysis is one of the important fields of mathematics. The focus of the first part of this book is on the presentation of the fundamental concepts in functional analysis: Banach space, linear operator, Hilbert space and dual space. The emphasis in the second part is on the main theorems in functional analysis and the third part is devoted to compact operators which play a crucial role in the theory of integral equations and partial differential equations and in mathematical physics.

Len Meas · Royal University of Phnom Penh · Date posted: September 11, 2022

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These notes cover standard undergraduate material from the theory of complex-valued functions, including the Cauchy–Riemann equations, Cauchy’s Theorem, the residue theorem, power series and Laurent series, the classification of singularities, and the Riemann sphere. Students are assumed to have studied multivariable calculus and to have some familiarity with set theory and proofs, but no prior experience in analysis is required. Some historical and cultural background is included in footnotes.

Joshua Bowman · Pepperdine University · Date posted: August 23, 2022 · Date revised: September 5, 2022

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